présentée en vue d obtention du délivré par l Université Toulouse III - Paul Sabatier Spécialité : Mathématiques Appliquées par Raymond EL HAJJ

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1 UNIVERSITÉ TOULOUSE III - PAUL SABATIER U.F.R Mathématiques Informatique Gestion THÈSE présentée en vue d obtention du DOCTORAT DE L UNIVERSITÉ DE TOULOUSE délivré par l Université Toulouse III - Paul Sabatier Spécialité : Mathématiques Appliquées par Raymond EL HAJJ intitulée Étude mathématique et numérique de modèles de transport : application à la spintronique soutenue le 3 septembre 28 devant le jury composé de : Naoufel Ben Abdallah Directeur de thèse Université Toulouse III-Paul Sabatier Abderrahmane Bendali Examinateur INSA de Toulouse Thierry Goudon Rapporteur INRIA Lille, Université de Lille 1 Ansgar Jüngel Rapporteur Université technique de Vienne Florian Méhats Examinateur Université de Rennes 1 Pierre Renucci Examinateur INSA de Toulouse Jean-Michel Roquejoffre Invité Université Toulouse III-Paul Sabatier Institut de Mathématiques de Toulouse Équipe Mathématiques pour l Industrie et la Physique (MIP) Unité Mixte de Recherche CNRS - UMR 5219 UFR MIG, Université Paul Sabatier Toulouse 3, 118 route de Narbonne, 3162 TOULOUSE cedex 9, France

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5 Remerciements Je tiens à remercier en premier lieu mon directeur de thèse Naoufel Ben Abdallah qui m a proposé un sujet de thèse original et moderne. Je le remercie chaleureusement pour sa confiance, ses conseils précieux et pour le temps qu il m a accordé malgré son emploi de temps surchargé avec la direction du laboratoire. Je tiens aussi à souligner sa gentillesse et ses qualités humaines qui m ont permis de réaliser ce travail et de faire mes premiers pas dans la recherche dans un environnement très agréable. Mes remerciements les plus respectueux vont à Thierry Goudon et Ansgar Yüngel qui m ont fait l honneur d être rapporteurs de ce travail. Qu ils trouvent ici l expression de ma profonde reconnaissance. J adresse ma sincère reconnaissance à Florian Méhats pour ses conseils et pour l intérêt qu il a toujours porté à mon travail depuis mon stage de DEA. Je suis très heureux de le compter parmi les membres de mon jury. Je tiens à remercier Abderrahmane Bendali, Pierre Renucci et Jean-Michel Roquejoffre pour l intérêt qu ils ont bien voulu accorder à ma thèse en acceptant de participer au jury. Les différentes réunions avec l équipe opto-électronique du Laboratoire de Physique et Chimie de Nano-Objets (LPCNO) à l INSA de Toulouse ont été d une importance considérable pour la réalisation de ce travail (surtout la partie principale sur la spintronique). Je remercie vivement Xavier Marie, Thierry Amand, Pierre Renucci et toute l équipe pour le temps qu ils nous ont accordés et pour les différentes discussions qui ont toujours été fructueuses. Un grand merci à tous les membres de l équipe MIP de l Institut de Mathématiques de Toulouse qui rendent l ambiance vraiment agréable et détendue. Merci à toi Christine pour ta gentillesse et ton sourire que tu as su garder malgré la surcharge de travail que tu avais avant de partir. Je remercie tous les collègues et amis : mon ancien co-bureau Marc pour ta bonne humeur et pour tes blagues que je te faisais 5

6 6 répéter plusieurs fois (merci d avoir libéré le 3 septembre pour être présent à ma soutenance) ; Michaël avec sa grande patience pour le foot, merci pour le petit ballon (c est un beau souvenir) qui nous a accompagnés durant ces années de thèse dans les couloirs du 2 ème étage ; et désolé Mounir, on t a dérangé avec les parties de foot ces derniers moments mais je t assure qu on ne faisait pas exprès...bon courage pour la suite ; l éternel Jean-Luc qui est toujours là pour nous rappeler l heure de la pause thé, bon courage aussi pour la fin de thèse. Je salue aussi et remercie : Laetitia, Laurent, Dominique, Benjamin, Tiphaine, Mélanie, Salvador, Clément, Ali, Sébastien, Antoine, Davuth, Aude et Olivier. Je n oublie pas les amis à l INSA : Elie, Abdelkader M. et Abdelkader T. J ai une pensée aussi pour les anciens : Claudia (merci pour les encouragements et pour l intérêt que tu portes à ce travail), Nicolas (merci pour le code), Mehdi, Raphaël... Enfin et surtout, tous mes remerciements vont pour mes parents, mes sœurs et mon frère pour leur amour et leur soutien constant. Je suis heureux que vous ayez fait le déplacement du Liban pour être à mes côtés le jour de ma soutenance. Merci pour tout à toi aussi Pascale et tes deux anges Etienne et My-Lihn.

7 Summary This thesis is decomposed into three parts. The main part is devoted to the study of spin polarized currents in semiconductor materials. An hierarchy of microscopic and macroscopic models are derived and analyzed. These models takes into account the spin relaxation and precession mechanisms acting on the spin dynamics in semiconductors. We have essentially two mechanisms : the spin-orbit coupling and the spin-flip interactions. We begin by presenting a semiclassical analysis (via the Wigner transformation) of the Schrödinger equation with spin-orbit hamiltonian. At kinetic level, the spinor Vlasov (or Boltzmann) equation is an equation of distribution function with 2 2 hermitian positive matrix value. Starting then from the spinor form of the Boltzmann equation with different spin-flip and non spin-flip collision operators and using diffusion asymptotic techniques, different continuum models are derived. We derive drift-diffusion, SHE and Energy-Transport models of two-components or spin-vector types with spin rotation and relaxation effects. Two numerical applications are then presented : the simulation of transistor with spin rotational effect and the study of spin accumulation effect in inhomogenous semiconductor interfaces. In the second part, the diffusion limit of the linear Boltzmann equation with a strong magnetic field is performed. The Larmor radius is supposed to be much smaller than the mean free path. The limiting equation is shown to be a diffusion equation in the parallel direction while in the orthogonal direction, the guiding center motion is obtained. The diffusion constant in the parallel direction is obtained through the study of a new collision operator obtained by averages of the original one. Moreover, a correction to the guiding center motion is derived. In the third part of this thesis, we are interested in the description of the confinement potential in two-dimensional electron gases. The stationary one dimensional Schrödinger Poisson system on a bounded interval is considered in the limit of a small Debye length (or small temperature). Electrons are supposed to be in a mixed state with the Boltzmann statistics. Using various reformulations of the system as convex minimization problems, we show that only the first energy level is asymptotically occupied. The electrostatic potential is shown to converge towards a boundary 7

8 8 layer potential with a profile computed by means of a half space Schrödinger Poisson system. Key words. Semiclassical analysis, Wigner transformation, spin-orbit hamiltonian, spinor Boltzmann equation, micro-macro limit, diffusion limit, moment method, entropy minimization, drift-diffusion, SHE, Energy-Transport, two-component models, Spin-FET, finite elements, Gummel iterations, guiding-center approximation, high magnetic field, convex minimization, min-max theorem, concentrationcompactness principle, boundary layer.

9 Table of contents Remerciements 3 Summary 6 Introduction 13 I. Modèles de transport en spintronique I.1 Introduction à la spintronique I.2 Description des modèles utilisés I.3 Résumé des résultats II. Diffusion et champs magnétiques forts II.1 Introduction et position du problème II.2 Résultats obtenus III. Confinement III.1 Motivation et description du problème III.2 Résultats obtenus III.3 Commentaires I Transport models for semiconductor spintronics 47 1 Semiclassical analysis Introduction Schrödinger equation with general spin-orbit Hamiltonian Analysis of the Schrödinger equation with spin-orbit term Semiclassical limit semiclassical limit and partially confining potential Introduction and main result Application : subband model with Rashba spin-orbit effect Proof of Theorem References

10 1 TABLE OF CONTENTS 2 Hierarchy of kinetic and macroscopic models Introduction Assumptions and notations Study of spinor Boltzmann type models Two-component models Decoherence limit Diffusion limit with strong spin-orbit coupling : two-component Drift-Diffusion model A general spin-vector Drift-Diffusion model Diffusion limit: formal approach Diffusion limit: the rigorous approach Maximum Principle (Proof of Theorem 2.5.3) SHE model Other fluid models for semiconductor spintronics Energy-Transport model Drift-Diffusion with Fermi-Dirac statistics References Numerical Applications Modelling and numerical implementation of spin-fet Introduction A coupled quantum/drift diffusion model with spin-orbit effect Setting of the problem and numerical results Spin accumulation in inhomogeneous semiconductor interfaces Presentation of the system Numerical results Spin accumulation and Rashba spin-orbit effect References II Diffusion and high magnetic fields Diffusion and guiding center approximation Introduction Setting of the problem and main results Scaling Notations Main results Analysis of the operator Q η Expansion of X η with respect to η

11 TABLE OF CONTENTS Expansion of X η z Expansion of X η Proof of the main theorems Proof of Theorem Proof of Proposition Proof of Theorem Concluding remarks References III Confinement High density Schrödinger-Poisson Introduction and main results Introduction Main results Remark on the scaling Notation and definitions Schrödinger Poisson system on a bounded domain Analysis of the limit problem Properties of the fundamental mode of the Schrödinger operator on [, + ) Proof of Theorem Convergence analysis Comments Fermi Dirac statistics Boundary conditions and higher dimension References A Appendix 25 A.1 Concentration-Compactness principle A.1.1 Proof of Lemma A.2 Pauli Matrices

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13 Introduction Ce travail de thèse comporte trois parties. La partie principale porte sur l étude mathématique et l analyse numérique des phénomènes de transport en spintronique. Deux autres travaux ont été menés en parallèle. Le premier concerne l étude de l asymptotique de diffusion et l approximation centre-guide de systèmes de particules en présence de champs magnétiques forts. Dans un autre travail, nous nous intéressons à la description du profil de potentiel de confinement dans des gaz d électrons bidimensionnels en étudiant une asymptotique forte densité du système Schrödinger-Poisson unidimensionnel stationnaire. Nous résumons maintenant chacune des trois parties. Nous nous intéressons dans la première partie de cette thèse au transport des courants polarisés en spin dans des matériaux à base de semi-conducteur. Le mécanisme essentiel pouvant agir sur l orientation du spin électronique dans les semi-conducteurs est ce que l on appelle le couplage spin-orbite. Lorsque la structure étudiée présente une absence de symétrie, le couplage spin-orbite se traduit par l apparition d un champ effectif faisant précesser (ou tourner) le vecteur spin pendant les vols libres des électrons. Dans les structures semi-conductrices, on a essentiellement le couplage spin-orbite de Rashba et celui de Dresselhauss. Le terme de Rashba apparaît dans des couches d accumulations à l interface entre deux hétérostructures et est due à la forte asymétrie du puits quantique dans lequel se confine le gaz d électrons bidimensionnel. Le couplage de Dresselhauus quant à lui résulte de l asymétrie présente dans certains structures cristallines. Nous dérivons et analysons une hiérarchie de modèles allant du niveau microscopique au niveau macroscopique en tenant compte des différents mécanismes de rotation et de relaxation du spin électronique dans les semi-conducteurs. Au niveau microscopique, l hamiltonien spin-orbite lié à l absence de symétrie se représente par la forme suivante H SO = α Ω(t, x, k) σ, où σ est le vecteur des matrices de Pauli (..5), x, k sont respectivement la position et le vecteur d onde d une particule (k i x ), α est l ordre du couplage et Ω représente le champ effectif. Dans un premier lieu, nous effectuons une ana- 13

14 14 INTRODUCTION lyse semi-classique, via la transformation de Wigner, de l équation de Schrödinger avec un hamiltonien spin-orbite. Suivant l ordre du couplage spin-orbite par rapport à la constante de Planck adimensionnée, nous dérivons des modèles cinétiques à deux composantes ou spinorielle (avec une fonction de distribution à valeur matricielle). Partant ensuite de la spinor forme de l équation de Boltzmann (avec différents opérateurs de collisions avec et sans renversement du vecteur spin) et par des techniques d asymptotiques de diffusion, nous dérivons et analysons plusieurs modèles macroscopiques. Ils sont de type dérive-diffusion, SHE, Energie-Transport, à deux composantes ou spinoriels conservant des effets de rotation et de relaxation du vecteur spin. Nous validons ensuite ces modèles par des cas tests numériques. Deux applications numériques sont présentées : la simulation d un transistor à effet de rotation de spin et l étude de l effet d accumulation de spin à l interface entre deux couches semi-conductrices différemment dopées. Cette partie de thèse donne lieu à deux articles en préparation [46, 47]. Mots clés : Analyse semi-classique, transformation de Wigner, hamiltonien spinorbite, équation de Boltzmann spinorielle, passage cinétique fluide, limite de diffusion, méthode des moments, minimisation d entropie, dérive-diffusion, SHE, Energie- Transport, modèles à deux composantes, Spin-FET, éléments finis, itérations Gummel. Dans un autre travail, nous considérons une équation cinétique de type Boltzmann linéaire dans des domaines où un champ magnétique fort est appliqué. La présence de ce dernier introduit de fortes oscillations et donc des difficultés pour les simulations numériques. Nous étudions la limite de diffusion en supposant que le champ magnétique est unidirectionnel et tend vers l infini. Le modèle obtenu est un modèle macroscopique (moins coûteux numériquement que le modèle cinétique). Il est constitué d une équation diffusive dans la direction parallèle au champ magnétique et d une dérive représentant l effet centre-guide en présence d un champ électrique dans la direction perpendiculaire. Le terme de diffusion contient des moyennes de giration de l opérateur de collisions utilisé. Nous prouvons la convergence en utilisant des techniques d entropie pour traiter le comportement diffusif, et en conjuguant par les rotations locales induites par le champ magnétique pour tenir compte des oscillations. Ce travail fait l objet d une publication [13]. Mots clés : limite de diffusion, approximation centre-guide, champ magnétique fort. Dans la troisième partie de cette thèse, nous étudions la limite faible longueur de Debye (ou faible température) du système de Schrödinger-Poisson unidimensionnel stationnaire sur un intervalle borné. Les électrons sont supposés dans un mélange d états avec une statistique de Boltzmann (ou de Fermi-Dirac). En utilisant différentes reformulations du système comme des problèmes de minimisation

15 INTRODUCTION 15 convexe, nous montrons qu asymptotiquement seul le premier niveau d énergie est occupé. Le potentiel électrostatique converge vers une couche limite avec un profil calculé à l aide d un système de Schrödinger-Poisson sur le demi axe réel. Cette partie est publiée dans SIAM-Multiscale Modeling and Simulations [48]. Mots clés : minimisation convexe, théorème min-max, principe de concentrationcompacité, couche limite. Dans la suite de cette introduction nous détaillons et présentons les principaux résultats obtenus dans chacune des trois parties de cette thèse. I. Modèles de transport en spintronique I.1 Introduction à la spintronique Les électrons ne sont pas seulement caractérisés par leur charge électrique mais aussi par leur moment cinétique intrinsèque ou spin. Jusqu aux années 9, l électronique ignorais quasiment le spin de l électron. La spintronique, ou l électronique de spin, est un nouveau domaine de recherche tentant d allier l électronique classique et les propriétés quantiques du spin. Il vise à manipuler le spin des porteurs de charge, et de l utiliser comme un degré de liberté supplémentaire ou comme un nouveau vecteur de l information. La magnétorésistance géante (Giant Magneto-Resistance ou GMR) découverte par Albert Fert et al [55], et la magnétorésistance tunnel, sont les premières manifestations de la spintronique. Dans des structures électroniques composées de couches magnétiques séparées par une couche paramagnétique, la GMR se traduit par un changement de résistance important observé dans de tels structures lorsque, sous l effet d un champ magnétique extérieur (ou sous l effet de l accumulation des spins à l interface M/PM), les aimantations macroscopiques des couches magnétiques successives basculent d un état antiparallèle à un état parallèle aligné. Un effet similaire à la magnétorésistance géante, appelé magnétorésistance tunnel, a été observé dans des jonctions tunnel métal/isolant/métal, dans lesquels les deux électrodes métalliques sont magnétiques. Cet effet magnétorésistif a été utilisé, dans les années quatre-vingt dix, pour développer des mémoires magnétiques à accès aléatoire ou MRAM (Magnetic Random Access Memories). Dans ces mémoires, l information n est plus stockée sous la forme d une charge, comme c est le cas des mémoires semi-conductrices de type DRAM ou Flash, mais sous la forme d une direction d aimantation dans la jonction tunnel magnétique. Pour utiliser le spin comme un porteur de l information, il faut que cette dernière

16 16 INTRODUCTION ne soit pas perdue durant son transport. En d autres termes, il faut disposer de porteurs dont l orientation du spin est parfaitement définie. Cette notion conduit naturellement aux courants polarisés en spin : essentiellement liée dans les semiconducteurs aux différences relatives des densités de spin-up et spin-down. Notons que dans les métaux ferromagnétiques, la notion de courants polarisés en spin est surtout liée à la différence de mobilité des spin-up et spin-down. Bien que les premières recherches dans ce domaine ont été menées pour des structures composées de multicouches magnétiques, les chercheurs portent actuellement une attention particulière à l étude des courants polarisés en spin dans les semi-conducteurs. La raison est la découverte du long temps de vie du spin et la présence des mécanismes pouvant agir sur la dynamique du spin électronique dans les semi-conducteurs. I.1.1 Mécanismes agissant sur le spin dans les semi-conducteurs Ces mécanismes sont dus au couplage spin-orbite qui est un effet relativiste lié au mouvement de l électron autour de son noyau. Ils peuvent être classés en deux catégories. Mécanisme d Elliot-Yafet. L interaction spin-orbite mélange les états de spin-up et down. Les intéractions instantanées entre les particules et le cristal (ou l environnement) peuvent alors être accompagnées d un retournement de l orientation du vecteur spin, selon le mécanisme dit d Elliot-Yafet [13, 56]. Bien que les intéractions avec renversement du spin soient des événements rares dans les semi-conducteurs [24], elles peuvent être suffisantes dans les zones à faible mobilité (ou forte densité) pour faire disparaître la cohérence en spin (ou faire relaxer le vecteur spin). C est le mécanisme de relaxation d Elliot-Yafet. Mécanisme de relaxation de D yakonov-perel. Lorsque le système présente une asymétrie d inversion, le couplage spin-orbite va se traduire par l apparition d un champ magnétique (qu on appelle champ effectif) faisant précesser le vecteur spin pendant les vols libres des porteurs de charge. Le couplage spin-orbite se décompose en deux termes : Couplage spin-orbite de Rashba [27]. Ce couplage apparaît dans les couches d accumulations à l interface entre deux hétéro-structures et dû à la forte asymétrie du puits quantique dans lequel se confine le gaz d électrons bidimensionnel (2DEG). Le vecteur de précession de spin associé au couplage de Rashba pour un 2DEG formé dans un plan (xy) est donné par [82, 27] : Ω R = 2a 46E z ( k y e x + k x e y ) (..1) avec a 46 est une constante dépendant du matériel, est la constante de Planck, E z est le champ électrique de confinement dans la direction z per-

17 INTRODUCTION 17 pendiculaire au plan (xy), k = (k x, k y ) est le vecteur d onde de particule dans le plan (xy) et e x, e y dénotent les vecteurs unitaires suivant les axes des x et des y. Couplage de Dresselhauss [43]. C est un couplage spin-orbite qui résulte de l asymétrie présente dans certaines structures cristallines. Le vecteur de précession de Dresselhauss s écrit sous la forme Ω D = 2a 42 (k xe x k y e y ), (..2) où a 42 est un paramètre dépendant de la structure. Ω R 1DEG S 1 S2 S 3 x S Ω R 2 S 3 Source z y S 1 Ω R Drain Plan 2DEG x Ω R Fig. 1 Dynamique du vecteur spin sous l action du vecteur de précession de Rashba dans un 2DEG et dans un fil quantique 1DEG. Le mécanisme de D yakonov-perel lié à l existence d un champ magnétique effectif représente le mécanisme essentiel de relaxation du spin électronique dans les hétéro-structures semi-conductrices [45, 13]. Le module du champ effectif lié au terme de Rashba (..1), soit la vitesse de rotation de spin, dépend de E z. Elle peut donc être contrôlée à l aide du champ électrique de confinement par un potentiel extérieur appliqué au système (potentiel de grille). Néanmoins, ce contrôle n est efficace que si l on se place dans de bonnes conditions. En effet, dans un gaz d électrons 2D, la direction du champ Ω R (..1) dépend du vecteur d onde k. Ce vecteur se redistribue de façon aléatoire dans le plan à l issu de chaque intéraction subie par la particule (voir Figure 1). Dans un régime fortement collisionnel, les particules subissent beaucoup de chocs. La dynamique du champ de Rashba ressemble dans ce cas à une marche au hasard et la cohérence de spin est donc relaxée via le mécanisme de relaxation de D yakonov-perel. Pour éviter ce problème, une solution consiste à confiner les électrons dans la direction y en plus du confinement dans la direction z. Le transport dans le gaz se fait alors dans une seule direction de l espace. On définit un fil quantique

18 18 INTRODUCTION ou 1DEG. Dans ce cas, le champ effectif de Rashba est donné par Ω 1D R = αk x e y, pour un certain paramètre α. Sa direction ne dépend pas de k et donc Ω 1D R ne change pas de direction avec les intéractions instantanées des particules. L effet de Rashba est dans ce cas efficace pour contrôler la dynamique du vecteur spin dans le fil quantique. Si le vecteur spin des électrons injectés dans le fil est parallèle à ce dernier, la rotation de spin s effectue dans un même plan perpendiculaire à Ω 1D R. La période de rotation varie avec la tension de grille appliquée. Ce mécanisme est vérifié numériquement, voir Chapitre 3 de cette thèse. Le vecteur de Dresselhauss, quant à lui, n est pas contrôlable par une voie externe et il induit aussi un mécanisme de relaxation de spin de type D yakonov Perel [45]. D autres mécanismes de relaxation de spin existent dans la littérature, voir [13]. I.1.2 Transistor à effet de rotation de spin Après le MRAM, la recherche actuelle se dirige vers la fabrication des composantes intégrant des matériaux magnétiques et semi-conducteurs dans une même hétéro-structure dite hybride. La possibilité de contrôler la vitesse de rotation du vecteur spin dans les hétéro-structures via le couplage spin-orbite de Rashba a conduit deux chercheurs américains Datta et Das à proposer en 199 [3] un transistor à effet de rotation de spin ou spin-fet (spin Field Effect Transistor). Il s agit d un transistor à haute mobilité électronique HEMT (High Electron Mobility Transistor) dans lequel les zones fortement dopées de source et de drain sont remplacées par des contacts ferromagnétiques. La source agit comme un polariseur en spin. Dans le semi-conducteur, les spins vont précesser autour d un certain champ effectif. Leur vitesse angulaire peut être modulée par la tension de la grille. Le contact de drain, quant à lui, agit comme un analyseur : si le spin est orienté parallèlement à l aimantation du drain, le courant dans ce dernier est important. Dans le cas contraire, le courant est faible. A part les contraintes de dimensionnement, plusieurs obstacles s opposent à la réalisation du spin-fet, notamment les problèmes d injection et de collection de spin aux interfaces FM/SC, SC/FM. I.2 Description des modèles utilisés Dans cette première partie de la thèse, nous nous intéressons à la dérivation des modèles de transport tenant compte de différents mécanismes agissant sur le spin

19 INTRODUCTION 19 électronique dans les semi-conducteurs décrits ci-dessus. Nous dérivons et analysons une hiérarchie de modèles allant du niveau microscopique (avec l équation de Schrödinger) au niveau macroscopique en passant par des modèles cinétiques (Chapitre 1, 2). Nous présentons ensuite quelques applications numériques (Chapitre 3). Dans cette section nous présentons les modèles utilisés pour décrire le transport des courants polarisés en spin dans les semi-conducteurs. Plus particulièrement, nous décrivons comment sont introduit dans les équations, pour différentes échelles de modélisation, les mécanismes de relaxation dûs aux couplages spin-orbite et aux intéractions avec renversement de spin. I.2.1 Modèle quantique En mécanique quantique, une particule de spin 1/2 (électron) plongée dans un potentiel V peut être décrite par une fonction d onde Ψ(t, x) = (ψ (t, x), ψ (t, x)) à valeur vectoriel dans C 2. Les composantes ψ (t, x) et ψ (t, x) représentent les fonctions d ondes des particules avec spin-up et spin-down respectivement. La fonction Ψ vérifie l équation de Schrödinger suivante : i t Ψ = (H + H SO )(Ψ) (..3) où H est l hamiltonien standard de l énergie cinétique plus l énergie potentiel H = ( 2 2m x + V )I 2, m est la masse effective d un electron et I 2 est la matrice identité de C 2. L hamiltonien spin-orbite, noté par H SO, s écrit sous la forme générale suivante : H SO = α Ω(t, x, k) σ (..4) où σ est le vecteur σ = (σ 1, σ 2, σ 3 ) des trois célèbres matrices de Pauli, ( ) ( ) ( ) 1 i 1 σ 1 =, σ 2 =, σ 3 = 1 i 1 (..5) et α est l ordre du couplage. Ici, k est le vecteur d onde, k i x. En effet, selon [76, 49] l hamitonien de l intéraction spin-orbite, dérivé de l équation de Dirac à quatre composantes [29], est donné par H SO = L hamiltonien de Rashba s écrit ([27]), i 2 4m 2 c 2 ( xv x ) σ. (..6) H R = αi ( σ 1 y σ 2 x )

20 2 INTRODUCTION pour un gas d électrons formé dans le plan (xy) et l hamiltonien de Dresselhauss est donné par [43] H D = αi ( σ 1 x σ 2 y ). Rappelons qu en physique quantique Ψ(t, x) 2 = ψ 2 + ψ 2 représente la probabilité de présence d une particule en x à l instant t et que l on a Ψ(t, x) 2 dx = 1. R 3 Les quantités macroscopiques telles que la matrice densité et courant sont définies à partir de Ψ comme suit N(t, x) = Ψ(t, x) Ψ(t, x), J(t, x) = 2i [ xψ(t, x) Ψ(t, x) Ψ(t, x) x Ψ(t, x)], où désigne le produit tensoriel de deux vecteurs. I.2.2 Modèles cinétiques et macroscopiques En microélectronique classique, le transport des charges est décrit au niveau cinétique par une grandeur statistique : la fonction de distribution scalaire f(t, x, v). Cette fonction représente une densité dans l espace des phases décrit par la position x et la vitesse v à l instant t. Autrement dit, f(t, x, v)dxdv correspond au nombre de particules se trouvant à l instant t dans un volume dxdv autour du point (x, v). L évolution de cette fonction est décrite par l équation de Vlasov ou Boltzmann dans un cadre collisionnel [22, 28, 65, 1]. En spintronique, un ensemble de particules de spin 1/2 est décrit au niveau cinétique par une fonction de distribution à valeurs dans l ensemble des matrices carrées hermitiennes d ordre 2 (H 2 (C)). L équation de Vlasov spinorielle avec intéractions spin-orbite s écrit t F + v x F + F v F = αi 2 [ Ω σ, F ], (..7) où F(t, x) = x V (t, x) est la force extérieure exercée sur les particules, et supposée conservative donc dérivant d un certain potentiel V. Dans cette description cinétique, le couplage spin-orbite est donné par le terme de droite de l équation (..7), où α est l ordre du couplage, Ω(t, x, v) est le champ effectif associé et [A, B] = AB BA désigne le commutateur des deux matrices. La fonction de distribution admet dans ce cas quatre degrés de liberté : un pour la distribution des charges et trois pour la distribution des spins. En effet, la matrice identité I 2 et les trois matrices de Pauli (..5) constituent une base de H 2 (C). Décomposons F dans cette base de la manière suivante F (t, x, v) = 1 2 f c(t, x, v)i 2 + f s (t, x, v) σ (..8) et injectons la dans (..7). Nous obtenons { t f c + v x f c x V v f c = t fs + v x fs x V v fs + α Ω f s =.

21 INTRODUCTION 21 On obtient une équation scalaire sur f c = tr(f ), où tr(f ) est la trace de F et f c représente la fonction de distribution des charges que l on utilise en microélectronique. La fonction f s est une fonction vectorielle à valeurs dans R 3. Elle représente la fonction de distribution des spins. Avec la décomposition (..8), le commutateur représentant l effet spin-orbite dans l équation (..7) devient un produit vectoriel entre Ω et f s. Ce dernier introduit un effet de rotation du vecteur distribution des spins f s autour du champ effectif Ω. De plus, les valeurs propres de F (t, x, v) pour tout (t, x, v) R + R 6 sont données par f (t, x, v) = 1 2 f c(t, x, v) + f s (t, x, v), f (t, x, v) = 1 2 f c(t, x, v) f s (t, x, v), où f s est le module de f s. Elles représentent les fonctions de distributions des particules avec spin-up et spin-down respectivement. Ceci montre que f c = f + f est la fonction de distribution de l ensemble total des particules sans tenir compte de leur spin (ou distribution des charges) et f s = 1 2 (f f ) est ce qu on appelle la fonction de distribution de polarisation en spin. Cette décomposition appliquée à toute quantité spinorielle (ou matricielle à valeur dans H 2 (C)) sera appelée décomposition en partie charge et partie spin. Si l on veut prendre en compte, en plus des intéractions spin-orbite, les collisions entre les particules (ou avec le crystal), avec et sans retournement de spin, l équation de Vlasov est alors remplacée par la spinor forme de l équation de Boltzmann (ou l équation de Boltzmann spinorielle) t F + v x F x V v F = Q(F ) τ + αi 2 [ Ω σ, F ] + Q sf (F ) où Q est l opérateur de collisions sans renversement de spin et τ est le temps moyen entre deux collisions successives. Les intéractions avec renversement de spin (ou spin-flip interactions) sont données par l opérateur Q sf admettant la forme réduite suivante Q sf = tr(f )I 2 F τ sf, (..9) où τ sf est le temps de relaxation du vecteur spin. Cette expression nous dit que si les intéractions avec retournement de spin sont nombreuses ou si τ sf est petit et tend vers zéro, alors la fonction de distribution F tend vers une distribution scalaire. Autrement dit, Q sf fait relaxer la distribution des spins donnée par la décomposition (..8) vers zéro (mécanisme de relaxation d Elliot-Yafet). Différents opérateurs de collisions sans renversement de spin seront considérés dans la suite comme les collisions électrons-phonons, élastiques, inélastiques, etc...

22 22 INTRODUCTION En microélectronique, les modèles macroscopiques ou fluides s intéressent à l évolution des quantités moyennées en vitesse de la fonction de distribution tels que la densité n(t, x) = f(t, x, v)dv, le courant j(t, x) = vf(t, x, v)dv, et l énergie R 3 R 3 W(t, x) = v 2 f(t, x, v)dv. Ces modèles moins précis que les modèles cinétiques R 3 2 (d un point de vue physique) possèdent une avantage d être en général moins coûteux du point de vue numérique. Différents modèles existent dans la littérature et sont obtenus à partir des modèles cinétiques par différents processus. Les modèles hydrodynamiques comme les équations d Euler et de Navier-Stokes sont obtenus avec une limite hydrodynamique reposant sur la méthode de moments [3, 22, 66, 71, 7, 99]. Une hiérarchie d autres modèles fluides existent tels que le modèle SHE (Spherical Harmonic Expansion) [32, 23], ET (Energie-Transport) [1, 12, 42, 36] et le modèle de dérive-diffusion [2, 95]. Ces différents modèles sont obtenus à partir de l équation de Boltzmann suivant le mécanisme collisionnel dominant [8] et via la limite de diffusion. Cette limite consiste à perturber la fonction de distribution autour d un équilibre thermodynamique local (la Maxwellienne) par un petit paramètre représentant le rapport entre le libre parcourt moyen et la longueur macroscopique caractéristique. Nous citons d autres travaux concernant l obtention rigoureuse des modèles macroscopiques à partir des équations cinétiques [2, 4, 4, 41, 67, 72, 73]. Modèles macroscopiques en spintronique. Les modèles macroscopiques (et cinétiques) utilisés pour décrire le transport des courants polarisés en spin sont de deux types. On a d une part les modèles à deux composantes et d autre part les modèles spinoriels ou matriciels. Dans la description à deux composantes, les électrons sont supposés avoir deux types de spin : électron avec spin-up et électron avec spin-down. Chaque type de particules est décrit par une équation cinétique ou macroscopique et les deux équations sont couplées par des termes d échanges dûs aux intéractions avec renversement de spin. Par exemple, le modèle de dérive diffusion à deux composantes s écrit [89, 11, 12] t n ( ) + div x j ( ) = n ( ) n ( ), τ sf j ( ) = D ( ) ( x n ( ) + x V n ( ) ), où n ( ) est la densité des particules avec spin-up (spin-down), j ( ) est le courant et D ( ) est la constante de diffusion correspondant au type de particules. Le terme de droite de cette équation est un terme de relaxation caractérisé par τ sf, temps de renversement de spin phénoménologique (spin-flip time) ou temps de relaxation, qui peut prendre en compte à la fois la relaxation via le mécanisme d Elliot-Yafet et la relaxation via le mécanisme de D yakonov-perel. Les modèles à deux composantes ont été initialement utilisés pour décrire le transport de spin dans les métaux ferromagnétiques. Ils sont ensuite utilisés dans les semi-conducteurs pour étudier par

23 INTRODUCTION 23 exemple la propagation d un courant polarisé en spin à travers l interface entre deux régions semi-conductrices de différents dopages [89] (voir aussi Chapitre 3). Dans ce type de modèles, l effet du mécanisme de D yakonov-perel est pris en compte de manière très simplifiée dans le temps phénoménologique τ sf. L approche spinorielle ou matricielle, qui permet d incorporer dans le modèle le mécanisme de D yakonov- Perel de manière microscopique, est une description plus générale du transport polarisé en spin dans les semi-conducteurs. Dans cette description, la variable de spin est une quantité vectorielle à valeurs dans R 3 et les différents mécanismes de relaxation et de rotation décrits auparavant peuvent être pris en compte comme on vient de l expliquer sur l équation de Boltzmann. I.3 Résumé des résultats Chapitre 1. Le chapitre 1 est consacré à l analyse semi-classique de l équation de Schrödinger avec hamiltonien spin-orbite. Dans la première partie de ce chapitre, nous étudions la limite semi-classique d une équation de Schrödinger de la forme avec iε t Ψ ε = ε2 2 xψ ε + V ε Ψ ε + αω W ε (t, x, iε x ) σψ ε, (..1) Ψ ε (t = ) = Ψ ε I, (..11) et ε est la constante de Planck adimensionnée. L hamiltonien spin-orbite est représenté par l opérateur de Weyl Ω W ε (t, x, iε x ) σ associé au symbol Ω ε (t, x, ξ) σ donné par Ω W ε (t, x, iε x ) σ(ψ) = 1 ( Ω (2π) 3 ε t, x + y ), εξ σ(ψ(y))e i(x y).ξ dξdy. 2 R 3 ξ R 3 y La limite semi-classique (ε ) conduit à de modèles cinétiques. Cette limite est étudiée en appliquant des résultats de convergence importants dans la théorie de la limite semi-classique via la transformée de Wigner [64, 8]. De nombreux résultats mathématiques concernant l étude de l équation de Schrödinger ainsi que sa limite semi-classique peuvent être trouvés dans la littérature [7, 11, 25, 26, 64, 8, 84, 85, 86]. En comparant l ordre du couplage spin-orbite α avec ε, deux cas sont étudiés. Si α est du même ordre que ε, alors la limite semi-classique conduit à la spinor forme de l équation de Vlasov. Plus précisément, la transformation de Wigner associée à Ψ ε et définie par W ε (Ψ ε, Ψ ε )(t, x, ξ) = (2π) 3 R 3 e iη ξ Ψ ε (t, x ε η 2 ) Ψε (t, x + ε η 2 )dη converge dans un certain sens (voir Chapitre 1) vers W. Si α = O(ε), W vérifie t W + ξ x W x V ξ W = i[w, Ω σ],

24 24 INTRODUCTION W (, x, ξ) = W I, où V, Ω et W I sont respectivement les limites de V ε, Ω ε et W ε (Ψ ε I, Ψε I ) quand ε. Par ailleurs, si α est supposé constant par rapport à ε (α = O(1)), on obtient à la limite un modèle cinétique à deux composantes avec un splitting entre les niveaux d énergie up et down d ordre Ω. La limite est vérifiée dans ce cas en dehors de l ensemble E des (t, x, ξ) dans R + R 6 x,ξ où Ω s annule { E = (t, x, ξ) R + R 6 / Ω(t, } x, ξ) =. En d autres termes, nous avons W (t, x, ξ) = 1 2 w Ω c(t, x, ξ)i 2 + w s (t, x, ξ) Ω σ pour tout (t, x, ξ) R + R 6 \ E où E est l adhérence de E. En plus, les valeurs propres de W, w = w c 2 + w s et w = w c 2 w s, satisfont t w + ξ λ x w x λ ξ w = sur (R + R 6 x,ξ ) \ E t w + ξ λ x w x λ ξ w = sur (R + R 6 x,ξ ) \ E. Les énergies totales up et down, λ and λ, sont respectivement données par λ (t, x, ξ) = ξ V + Ω, λ (t, x, ξ) = ξ V Ω. Lorsque les courbes caractéristiques atteignent E, les deux modes d énergies λ et λ se croisent et un problème de transfert d énergie entre eux peut apparaître. Plusieurs travaux mathématiques existent pour décrire l évolution semi-classique d un système au delà d un croisement de modes et pour quantifier le transfert d énergie en termes de mesures de Wigner à double échelle et formule de Landau-Zener. Nous renvoyons le lecteur aux travaux de Patrick Gérard et Clotilde Fermanian-Kammerer sur ce sujet [5, 51, 52, 53, 54]. La deuxième partie du premier chapitre est consacrée à la dérivation d un modèle de sous-bande couplé cinétique/quantique. Ce type de modèle décrit le transport des particules dans des systèmes partiellement confinés tels que les gaz d électrons bidimensionnels. Dans ces systèmes de particules, les différentes directions de l espace ne jouent pas le même rôle. Le gaz d électrons est confiné dans une (ou plusieurs directions) et le transport s effectue dans les autres directions. Dans la direction du confinement l échelle spatiale est généralement petite et une description quantique est nécessaire ; dans la direction du transport les électrons se comportent d une façon classique et un modèle cinétique ou fluide peut être utilisé. Cette approche de couplage directionnel a été récemment développée au sein de l équipe MIP de l institut de mathématiques de Toulouse. Dans [15], un modèle de sous-bande quantique/cinétique est dérivé d un modèle entièrement quantique par le biais d une

25 INTRODUCTION 25 limite semi-classique partielle. L analyse du modèle obtenu a été ensuite effectuée dans [14]. La thèse de N. Vauchelet dirigée par N. Ben Abdallah et F. Méhats [98] a été consacrée à la dérivation et l étude mathématique et numérique d un modèle couplé quantique/fluide à savoir Dérive-Diffusion-Schrödinger-Poisson (voir aussi [17, 19, 9]). D autres modèles adiabatiques quantique/fluide sont dérivés comme les modèles Schrödinger/SHE et Schrödinger/ET [18]. Notons enfin qu une autre stratégie de couplage quantique/classique à savoir le couplage spatial existe [6, 9, 34]. Cette approche consiste à découper le domaine d étude en plusieurs zones. Chaque zone est décrite par un modèle quantique ou classique et le couplage se fait par des conditions d interfaces. Ici, le confinement a lieu dans une seule direction de l espace notée z et le transport s effectue dans la direction orthogonale x. Nous étudions la limite semi-classique partielle de l équation de Schrödinger avec terme spin-orbite en suivant [15]. Le point de départ est l équation de Schrödinger adimensionnée suivante iε t Ψ ε = ε2 2 xψ ε zψ ε + V ε Ψ ε + εω W ε (t, x, z, i ε ) σ(ψ ε ), (..12) Ψ ε (, x, z) = Ψ ε I (x, z), avec (x, z) R 2 [, 1], Ψ ε = Ψ ε (t, x, z) et ε = (ε x, z ). Les sous bandes sont définies par les éléments propres de l opérateur de Schrödinger z + V ε dans la direction z zχ ε p + V ε χ ε p = ɛ ε pχ ε p χ ε p(t, x,.) H 1 (, 1), 1 χ ε pχ ε q = δ pq. (..13) La limite ε, conduit à un modèle de sous bande quantique/cinétique. Plus précisément, la matrice densité donnée par N ε = Ψ ε (t, x, z) Ψ ε (t, x, z) converge dans un certain sens vers N avec N(t, x, z) = ( ) F p (t, x, ξ x )dξ x χ p (t, x, z) 2 p 1 R 2 où χ p sont les fonctions propres pour ε = et ξ x est la variable dual associée à x. La fonction de distribution de la pième sous bande satisfait t F p + ξ x x F p x ɛ p ξx F p = i[f p, Ω p σ], + conditions initiales où le champ effectif du pième sous bande, Ω p, est donné par Ω p (t, x, ξ x ) = 1 Ω(t, x, z, ξ x ) χ p (t, x, z) 2 dz. (..14)

26 26 INTRODUCTION En appliquant ces résultats à l hamiltonien spin-orbite (..6), l équation cinétique de la pième sous bande s écrit alors avec t F p + ξ x x F p x ɛ p ξx F p = iα p (t, x)[σ 2 v 1 σ 1 v 2, F p ] α p (t, x) = 1 z V (t, x, z) χ p (t, x, z) 2 dz. L effet spin-orbite obtenu n est autre que l effet de Rashba utilisé dans la littérature pour modéliser le couplage spin-orbite dans les 2DEG [13, 27]. De plus, nous obtenons une relation explicite reliant l ordre du couplage α p et le potentiel de confinement. Chapitre 2. Dans ce chapitre, nous dérivons rigoureusement une hiérarchie de modèles macroscopiques vectoriels ou à deux composantes. Nous partons de l équation de Boltzmann adimensionnée dans un scaling de diffusion F ε t + 1 ε (v. xf ε x V. v F ε ) = 1 ε Q(F ε )+ α [ ] i Ω(x, 2 ε 2 v). σ, F ε +Q sf (F ε ), (..15) où ε est le libre parcourt moyen adimensionné. Nous rappelons que Q sf est l opérateur de collisions avec renversement de spin donné par (..9). Différents opérateurs de collisions Q sans retournement de spin sont considérés. Suivant le mécanisme collisionnel dominant, nous dérivons de modèles macroscopiques de type dérive diffusion avec statistique de Boltzmann ou de Fermi-Dirac, SHE, ET, à deux composantes ou vectoriels gardant des effets de rotations et de relaxation du spin électronique. Nous commençons par considérer l approximation BGK linéaire de l opérateur de collisions électron-phonon avec une statistique de Boltzmann donnée par Q(F ) = α(v, v )[M(v)F (v ) M(v )F (v)]dv, (..16) R 3 où α est la section efficace supposée bornée par des bornes supérieures et inférieures strictement positives. La fonction M est la maxwellienne M(v) = 1 e 1 (2π) 3 2 v 2, v R 3. (..17) 2 Cet opérateur fait relaxer F vers la maxwellienne quand le libre parcourt moyen tend vers zero. Puisque les collisions avec renversement de spin sont rares dans les semiconducteurs [24], nous supposons que Q sf est une perturbation d ordre ε de Q. Ceci justifie que Q sf est considéré d ordre un dans le scaling de diffusion (..15). Nous étudions la limite de diffusion ε pour différents ordres du couplage spin-orbite α par rapport à ε. Le chapitre 2 est organisé de la manière suivante.

27 INTRODUCTION 27 Nous nous intéressons tout d abord à l étude des propriétés fondamentales de l équation de Boltzmann avec terme spin-orbite (..15). Nous montrons l existence et l unicité de solutions faibles vérifiant le principe de maximum. Autrement dit nous montrons que si initialement F ε (t =, x, v) := F in (x, v) est une fonction à valeurs dans l ensemble des matrices carrées hermitiennes et positives que l on note par H + 2 (C), alors F ε (t, x, v) H + 2 (C), t > et (x, v) R 6. Nous établissons en plus des estimations à priori nécessaires pour passer à la limite. La deuxième partie est consacrée à la dérivation des modèles cinétiques et macroscopiques à deux composantes. Nous nous plaçons dans un régime où le couplage spin-orbite est fort de sorte que la période de rotation du vecteur spin T autour de Ω est petite devant le temps caractéristique t. La limite η := T est appelée t limite de decohérence. Cette limite fait relaxer la partie spin de la fonction de distribution parallèlement au champ effectif considéré Ω(t, x, v). Dans le cas où la direction de Ω ne dépend pas de v, nous obtenons un modèle cinétique à deux composantes en projetant le vecteur distribution de spin suivant la direction de Ω(t, x,.). Partant ensuite de ce dernier modèle cinétique et appliquant une limite de diffusion (ou fluide), nous pouvons dériver des modèles macroscopiques à deux composantes. Par ailleurs, si Ω change de direction avec la vitesse v, une limite de décohérence appliquée à la spinor forme de l équation de Boltzmann suivie d une limite de diffusion fait relaxer la distribution de spin vers et on obtient à la limite un modèle macroscopique scalaire sur la densité des charges. Notons que cette relaxation de spin est bien connue en spintronique des semi-conducteurs et n est autre que le mécanisme de relaxation de D yakonov-perel [45, 13] présenté au début de ce chapitre introductoire. Ces résultats sont ensuite vérifiés rigoureusement en passant à la limite ε dans l équation (..15) en présence d un couplage spin-orbite ultra-fort : supposant α = O( 1 ). Nous montrons aussi que la limite de diffusion dans ce cas aboutit à un ε Ω modèle diffusif à 2 composantes si Ω ne dépend pas de v et à un modèle scalaire sinon. Section 2.5 concerne la dérivation d un modèle vectoriel de type dérive-diffusion avec des effets de rotation et de relaxation du vecteur spin. Remarquons tout d abord que si les intéractions spin-orbite sont faibles telles que α = O(ε) alors, formellement F ε F lorsque ε tend vers avec Q(F ) =. Ceci implique que F = N(t, x)m(v) (voir Chapitre 2). En plus, en intégrant l équation (..15) par rapport à v et passant à la limite, la matrice densité N vérifie t N + div x (D( x N + x V N)) = i 2 [ H e σ, N] + Q sf (N) où D est une matrice symétrique définie positive. Le champ effectif résultant de cette

28 28 INTRODUCTION limite est donné par H e (x) = Ω(x, v)m(v)dv. R 3 En pratique, Ω est impaire par rapport à v (voir vecteur de Rashba et de Dresselhauss par exemple). Ceci implique, puisque M est paire par rapport à v, que H e = et on perd l effet spin-orbite à la limite dans ce cas. Ainsi, pour garder de traces de l effet spin-orbite au niveau macroscopique, on étudie la limite de diffusion en supposant que le terme spin-orbite est d ordre 1 ε (α = O(1)) si Ω est impaire par rapport à v. Nous dérivons un modèle vectoriel de type dérive-diffusion conservant des effets de rotation et de relaxation du vecteur spin et l un de ces principaux résultats à savoir le principe de maximum est vérifié. En suivant la même stratégie qu on vient de présenter, d autres modèles macroscopiques à deux composantes ou matriciels peuvent être dérivés. Nous présentons à la fin du 2ème chapitre un modèle de type SHE dans le cas où les collisions prises en comptes sont les collisions élastiques. Nous discutons ensuite la construction des opérateurs de collisions non linéaires conservant certains moments tels que la masse et l énergie par le principe de minimisation d entropie. Nous utilisons ensuite ces opérateurs pour dérivés des modèles de type Energie-Transport et Dérive-Diffusion avec une statistique de Fermi-Dirac par le bias de la méthode des moments. Chapitre 3. Dans ce chapitre, deux applications numériques sont présentées. La première concerne la simulation d un transistor à effet de rotation de spin. En suivant le travail de [19, 9], un modèle de sous-bande de dérive diffusion Schrödinger-Poisson avec effets de relaxation et de rotation du vecteur spin dus au couplage de Rashba est dérivé et utilisé pour la simulation. Le dispositif considéré est un MOSFET à double grille (voir [98] pour plus de détails). Nous considérons un cas simple en supposant que le transport s effectue dans une seule direction de l espace. Le vecteur de precession de Rashba ne change pas de direction dans ce cas. Nous ne considérons pas des contacts ohmic ferromagnétiques. Nous injectons un courant polarisé en spin dans le plan du dispositif avec une densité de spin parallèle à la direction du transport. Nous montrons numériquement l efficacité de l effet de Rashba pour contrôler l orientation du vecteur spin dans le canal. La direction de spin à l arrivé au drain est entièrement déterminée par le potentiel de grille V gs et le courant de drain oscille en fonction de V gs dans ce cas. Le deuxième exemple étudié est l effet de l accumulation de spin à l interface entre deux régions semi-conductrices. C est un effet bien connu en spintronique des semi-conducteurs [89, 88]. Deux modèles sont utilisés. Cet effet sera représenté tout d abord en utilisant un modèle de dérive-diffusion à deux composantes couplé avec l équation de Poisson. Nous utilisons ensuite un modèle vectoriel de dérive diffusion

29 INTRODUCTION 29 couplé toujours avec l équation de Poisson afin d étudier l effet de precession de Rashba sur la densité d accumulation.

30 3 INTRODUCTION II. Diffusion et champs magnétiques forts (Chapter 4) II.1 Introduction et position du problème Le confinement magnétique est une approche importante utilisée dans beaucoup de dispositifs de production d énergie par fusion nucléaire (tokamaks, plasmas ionosphériques, etc...). De puissants champs magnétiques sont utilisés dans de tels dispositifs. Le confinement est basé sur la propriété des particules de décrire une trajectoire en hélice autour d une ligne de champ magnétique. En effet, le mouvement d une particule de masse m et de charge q plongée dans un champ électrique E et un champ magnétique constante B = Be z, où e z est un vecteur unitaire, est décrit par l équation de mouvement suivante : m dv dt = q( E + B(v e z )), (..18) où v est le vecteur vitesse. La solution générale de l équation homogène associée (pour E = ) est donnée par v(t) = R(w c t)v avec R(τ) = cos τ sin τ sin τ cos τ 1 (..19) où w c = qb m est la fréquence du cyclotron et v R 3. Une quantité importante prise en considération dans la suite est la gyropériode T c. Il s agit du temps qu elle met une particule de masse m et de charge q soumise à un champ magnétique constant B pour faire une rotation d angle 2π. On a T c = 2π w c = 2πm qb. (..2) En intégrant l équation (..19), le rayon vecteur x(t), ẋ(t) = v(t), vérifie x (t) = x () + 1 (e z v ) v R(w c t + π w c w c 2 ) v v x (t) = x () + v t, où x et x sont respectivement la partie perpendiculaire et la partie parallèle de x par rapport au champ magnétique ( (ou e z ). C est l équation paramétrée d une hélice de centre de rotation x c (t) = x () + 1 ) (e z v ), x () + v w t qu on ap- c pelle centre-guide et de rayon de rotation r L = v w c = m v qb, (..21)

31 INTRODUCTION 31 appelé rayon de Larmor et qui est inversement proportionnel au module du champ magnétique B. Ceci implique que, lorsque B augmente et tend vers l infini, le rayon de Larmor tend vers zéro et les particules sont piégées le long de la ligne du champ magnétique. En présence d un champ électrique, la solution de (..18) est donnée par la solution générale de l équation homogène v plus une solution particulière v part de (..18). On a v part = q E t et on cherche une solution particulière stationnaire m dans la direction perpendiculaire. Ceci implique que B vpart = E et donc vpart E = B. (..22) B 2 On en déduit que le rayon vecteur en présence d un champ électrique est donné par x (t) = x () + 1 w c (e z v E ) + B t r B 2 L R(w c t + π 2 ) v v x (t) = x () + (v + q E m )t. Par conséquent, un champ électrique perpendiculaire à B n accélère pas les particules mais crée une dérive uniforme du centre-guide dans la direction E B avec une vitesse (appelée vitesse de dérive du centre-guide) inversement proportionnelle à B 2, et donnée exactement par (..22). La présence d un champ magnétique fort dans les équations cinétiques (Vlasov, Boltzmann) introduit de fortes oscillations et donc des difficultés pour la simulation numérique. La question de trouver des modèles approximatifs moins coûteux numériquement que les modèles cinétiques est très importante dans ce sujet. Différents modèles approximatifs existent dans la littérature tels que les modèles centre-guide et gyrocinétiques. Ces modèles consistent à moyenner le mouvement sur la gyropériode tout en supposant que B tend vers l infini. Nous référons à une liste de travaux physiques sur ce type de modèles [77, 81, 87, 44]. De point de vue mathématique, E. Sonnendrücker et E. Frénod ont étudié la limite champ magnétique fort du système Vlasov et Vlasov-Poisson par des techniques d homogénéisation [59, 6]. Dans [59], il a été montré que l approximation centre-guide de l équation de Vlasov 3D conduit à une équation cinétique unidimensionnelle dans la direction du champ magnétique. La dérive du centre-guide est obtenue ensuite dans [6] en étudiant la limite centreguide de l équation de Vlasov 2D (dans la direction perpendiculaire au champ) dans un échelle de temps suffisamment long. L investigation de l équation de Vlasov et du système Vlasov-Poisson en présence d un champ magnétique fort est intensivement étudiée pour différents régimes asymptotiques [57, 58, 61, 62, 68, 69, 96, 97]. Dans toutes les références précédentes, le transport est supposé balistique (sans collisions). Nous sommes intéressés ici par des régimes où les collisions sont importantes et nous étudions l asymptotique de diffusion de l équation de Boltzmann en

32 32 INTRODUCTION présence d un champ magnétique fort. L équation de Boltzmann dans le scaling de diffusion s écrit [95, 67] ε t f ε + (v r f ε + E v f ε ) + α(v e z ) v f ε = Q(f ε) ε où r = (x, y, z) est le vecteur position, v = (v x, v y, v z ) est la variable vitesse, et ε = τ 1 est le libre parcourt moyen adimensionné avec t et le temps caractéristique. t Le champ magnétique est supposé constant et parallèle à l axe des z de vecteur unitaire e z. Nous notons par r = (x, y) et v = (v x, v y ) les variables orthogonales. Le paramètre α 1 représente la gyropériode adimensionnée où α = tw c = 2πt T c. Nous considérons dans ce travail pour simplifier la forme BGK linéaire de l opérateur des collisions electron-phonon donné par (..16)-(..17). Le modèle fluide obtenu lorsque ε tend vers zéro est alors de type dérive-diffusion. Si la gyropériode est plus grande que le temps de relaxation τ ou si α est constante devant ε (α = O(1)) alors l effet du champ magnétique disparaît à la limite. D autre part, si T c est du même ordre que le temps de relaxation ou bien si α = O(ε 1 ), l équation de Boltzmann devient t f B ε + 1 ε (v rf B ε + E v f B ε ) + B ε 2 (v e z) v f B ε = Q(f B ε ) ε 2 (..23) où B est une constante strictement positive telle que α = B. Ce scaling de diffusion ε est utilisé généralement pour les plasmas et les gas binaires. La matrice de diffusion dans le modèle limite admet une composante antisymétrique dans ce cas générée par le champ magnétique. Ce résultat est prouvé par P. Degond et al [31, 35, 39] pour des collisions avec des murs. D autres résultats formels concernant les gaz binaires se trouvent aussi dans [33, 37, 38, 83]. Dans ce travail, nous considérons une situation où la gyropériode T c est plus petite que le temps de relaxation. Ce qui revient à supposer que B est grand et tend vers l infini. Nous allons voir que le scaling de diffusion standard (..23) n est pas convenable pour décrire, quand ε et B +, la dynamique du centreguide dans la direction perpendiculaire au champ magnétique. Il décrit seulement le transport diffusif dans la direction parallèle. Un autre scaling est proposé pour lequel la limite de diffusion avec B + conduit à un modèle de dérive-diffusion dans la direction parallèle au champ magnétique et dans la direction perpendiculaire, le transport est dominé par la dérive du centre-guide. Afin de mieux comprendre ce qui se passe à la limite, nous considérons l approximation du temps de relaxation de l opérateur de collisions Q(f) = n(t, x)m(v) f, n(t, x) = f(t, x, v)dv. R 3 (..24)

33 INTRODUCTION 33 En prenant un développement de Hilbert de f B ε de la forme f B ε = f B + εf B 1 + O(ε 2 ) et en injectant dans (..23), on trouve { Q(f B ) B(v e z ) v f B = Q(f B 1 ) B(v e z ) v f B 1 = v r f B + E v f B. La première équation implique que f B devient = n B (t, r)m(v) et la deuxième équation f B 1 B(v e z ) v f B 1 = ( r n B E n B )vm. (..25) En intégrant (..23) par rapport à v et en passant à la limite, nous obtenons l équation de continuité t n B + div r j B = où j B = R 3 vf B 1 dv est le courant. D autre part, avec (..25) j B se calcule explicitement et nous avons j B = D B ( r n B E n B ) où D B = 1 B 1+B 2 1+B 2 B 1+B B 2 1. La matrice de diffusion D B n est pas symétrique. En plus, D B 1 D S 1+B 2 1 B = 1+B 2 1 = D S B + DAS B avec est une matrice symétrique définie positive avec différents coefficients de diffusion suivant la direction parallèle et la direction perpendiculaire. La partie antisymétrique est donnée par : D AS B = B B 2 1. L équation limite s écrit alors t n B div r ( D S B ( r n B E n B ) ) + B 1 + B 2 (E e z). r n B =. La partie antisymétique de la matrice de diffusion donnant l effet centre guide est d ordre 1. Ainsi quand B tend vers l infini, nous obtenons une équation de dérivediffusion 1D dans la direction z avec ce B scaling. Puisque le champ magnétique confine les particules dans la direction perpendiculaire r = (x, y), et afin de capter la dynamique du centre-guide dans cette direction, nous proposons de faire le changement de variable : r ( B)r. Ce rééchelonnement implique les changements suivants ( ) ( ) B r BE r, E. z E z

34 34 INTRODUCTION Pour simplifier la présentation, nous posons B = 1 où η est un petit paramètre η 2 destiné à tendre vers zéro. Dans ce travail, la fonction de distribution dépend de deux petits paramètres ε et η. Elle vérifie l équation de Boltzmann adimensionnée suivante f εη + T zf εη + T f εη t ε εη f εη (t = ) = f (r, v). + (v e z) ε 2 η 2 v f εη = Q(f εη ) ε 2 (..26) Ici, T and T z sont respectivement les parties perpendiculaire et parallèle de l opérateur de transport T = v r + E v, T z = v z z + E z vz. II.2 Résultats obtenus Nous étudions le cas où ε tend vers zéro pour η donné et ensuite η tend vers zéro et le cas où ε et η tendent vers zéro simultanément. Le premier résultat concerne la limite de diffusion ε pour η fixé. Nous montrons que (f εη ) ε converge dans un certain sens (voir Chapitre 4) vers ρ η (t, r)m(v) et la densité ρ η vérifie t ρ η div(d η ( ρ η ρ η E)) =. (..27) La matrice de diffusion D η est donnée par la formule ( ) ( ) 1 D η η = v ηx η dv, v z Xz η X η = Xη X η z étant l unique solution de Q(X η )+ v e z η 2 v (X η ) = 1 η 2 v v z M(v) and R 3 X η (v)dv =. (..28) Dans l approximation du temps de relaxation σ(v, v ) = 1 τ, la matrice Dη se calcule explicitement D η = τ η 2 τ τ 2 +η 4 τ 2 +η 4 η 2 τ 2 +η 4 τ 2 +η 4 1 À la limite η, le bloc supérieure de cette matrice se réduit à I = 1 1 et sa partie symétrique est d ordre η 2. Nous obtenons ces résultats dans le cas où.

35 INTRODUCTION 35 la section σ est non constante. Nous développons rigoureusement X η par rapport à η jusqu à l ordre η 4. Ceci nous donne un développement des parties symétrique et antisymétrique de D η en fonction de η. Nous montrons que la diffusion est d ordre 1 dans la direction parallèle et d ordre η 2 dans la direction perpendiculaire. La partie antisymétrique D η as agit dans la direction perpendiculaire. Elle est donnée par la matrice I à l ordre principale ce qui conduit à la dérive du centre-guide. En plus, le développement de D η as donne une correction d ordre η à la dynamique du centreguide. Nous renvoyons le lecteur au Chapitre 4 pour plus de détails. Le deuxième résultat concerne la limite simultanée ε, η. Nous montrons que f εη ρ(t, r)m(v) avec Le courant parallèle est donné par t ρ + z J z + r (ρe e z ) = ρ(, r) = f (r, v) dv R 3 (..29) J z = D z ( z ρ E z ρ) où D z est la constante de diffusion D z = X () z R 3 v z dv, et X () z est le terme d ordre zéro de Xz η. C est une fonction isotrope (invariante par rotation autour de e z ) vérifiant avec et Q(X () z ) = v z M, R 3 X () z (v)dv = Q(f)(v) = σ(v, v )[M(v)f(v ) M(v )f(v)]dv R 3 σ (v, v ) = 1 2π 2π σ (R(τ)v, R(τ )v ) dτdτ. 4π 2 L équation limite (..29) est composée d un courant de dérive-diffusion dans la direction parallèle et d une dérive due au mouvement du centre-guide dans la direction perpendiculaire. Le coefficient de diffusion parallèle D z est obtenu via l analyse d un opérateur de collisions avec une section efficace moyennée sur les cercles de gyration autour du champ magnétique.

36 36 INTRODUCTION III. Confinement (Chapitre 5) III.1 Motivation et description du problème Afin de mieux contrôler le transport électronique, de nombreux dispositifs électroniques sont basés sur la réduction de la dimensionalité en confinant les électrons dans une ou plusieurs directions de l espace. Parmi eux, nous avons les gas d électrons bidimensionnels (2DEG) [1, 74] dont les électrons sont confinés dans une seule direction. Le système Schrödinger-Poisson est l un des modèles les plus appropriés pour décrire le transport quantique balistique (sans collisions) des particules chargés dans les semi-conducteurs ainsi qu en chimie quantique. Dans [94, 93, 92], la simulation du transport électronique dans les gaz d électrons bidimensionnels balistiques a été réalisée grâce à la dérivation d un modèle approché permettant de tenir compte de manière moyennée de l effet du potentiel de confinement sur le transport quantique. Le modèle proposé admet un avantage considérable d être moins coûteux numériquement et de rester en accord complet avec le model 3D de point de vue physique. L analyse rigoureuse de cette approximation a ensuite été effectuée dans [16, 91] dans le cas où le profil du potentiel de confinement V c et l échelle spatiale de ses variations ε sont supposés donnés. Dans [16], l approximation est justifiée en effectuant une étude asymptotique lorsque ε tend vers zéro de la solution du système de Schrödinger-Poisson tridimensionnel suivant i t ψ ε = 1 2 x,zψ ε + V ε ψ ε + V ε c ψ ε, V ε = 1 4π x 2 + z 2 ( ψε 2 ), où x R 2 et z R est la direction de confinement. Le potentiel V ε est le potentiel auto-consistant et Vc ε par est le potentiel de confinement imposé au système et donné Vc ε = 1 ε V c( z ), (..3) 2 ε où V c est une fonction donnée. Le cas stationnaire dans un domaine borné est traité dans [91]. Le but de ce travail est en quelque sorte de justifier le scaling (..3) par l analyse d un système de Schrödinger-Poisson unidimensionnel (dans la direction de confinement z). De ce fait, nous ignorons le transport des particules dans la direction orthogonale à l axe des z en supposant par exemple que le système est invariant dans cette direction. Le paramètre ε n est pas donné mais calculé en fonction de la longueur de Debye adimensionnée. Le système est supposé fermé et occupe l intervalle [, 1]. Les

37 INTRODUCTION 37 électrons se répartissent dans ce cas sur des niveaux d énergies discrets qui sont les valeurs propres de l opérateur de Schrödinger d2 dz 2 + V. La densité totale de particules est la superposition des densités de tous les niveaux d énergies avec un facteur d occupation donné par la statistique de Boltzmann. Le potentiel electrostatique de confinement est noté par V ε. Le point de départ est le système Schrödinger-Poisson unidimensionnel stationnaire vérifié par V ε suivant d2 ϕ p dz 2 + V ϕ p = E p ϕ p z [, 1], ϕ p H 1 (, 1), ϕ p () =, ϕ p (1) =, = 1 e Ep ϕ p 2, Z = Z p=1 dv V () =, (1) =. dz ε 3 d2 V dz e Ep p=1 ϕ p ϕ q = δ pq (..31) Le scaling menant à ce système sera détaillé. Le paramètre ε est donné en fonction de la longueur de Debye adimensionnée, λ D, par ε 3 = ( λ D L )2, λ D = kb T ε ε r e 2 N (..32) avec L est la longueur caractéristique dans la direction z, e est la charge élémentaire, et N = N s L est la densité volumique moyenne avec N s le nombre total de particules (ou la densité surfacique). Le paramètre ε est supposé petit et tend vers zéro. Ceci correspond à la limite faible longueur de Debye ou faible température ou forte densité (vu la relation (..32)). A la limite (ε ), les fonctions d ondes se concentrent au point z =. Afin d analyser cette couche limite, un zoom est effectué au voisinage de z = via le changement de variables suivant ϕ p (z) = 1 ε ψ p ( z ε ), E p = 1 ε 2 E p, V (z) = 1 ε 2 U(z ε ), ξ = z ε. (..33) L équation de Poisson dans (..31) devient d2 U dξ 2 + p=1 = 1 Z + p=1 e E p ε 2 ψ p 2 avec Z = e Ep ε 2. Les valeurs propres (E p ) p sont simples, distinctes et forment une suite croissante par rapport à p pour tout ε >. Nous montrerons de plus l existence d un gap strictement positif séparant le premier niveau d énergie E 1 et les autres E p (p 2) uniformément par rapport à ε. Ceci implique que les facteurs d occupations e E p ε 2 pour p 2 sont tous négligeables devant le premier (p = 1) quand ε devient petit. Il est donc naturel de prétendre que le modèle (..31) est asymptotiquement proche

38 38 INTRODUCTION de d2 ϕ 1 dz + Ṽ ϕ 2 1 = Ẽ1 ϕ 1 z [, 1], { 1 1 } Ẽ 1 = inf ϕ 2 + Ṽ ϕ 2, ϕ H 1(,1), ϕ L 2 =1 (..34) ε 3 d2 Ṽ dz = ϕ 1 2, 2 dṽ Ṽ () =, dz (1) = dans lequel seul le premier niveau d énergie est pris en compte. Ceci est en accord avec les simulations numériques [94]. Dans ce travail, nous montrons rigoureusement que ces deux modèles sont asymptotiquement proches et nous estimons leur différence en fonction de ε. Nous montrons qu elle décroît exponentiellement. De plus, nous prouvons qu à la limite ε, le potentiel Ṽε solution de (..34) converge vers un potentiel de couche limite de la forme 1 ε 2 U (. ε ) avec un profil U vérifiant un système de Schrödinger-Poisson unidimensionnel sur le demi axe réel : d2 ψ 1 dξ + Uψ 2 1 = E 1 ψ 1 ξ [, + [, { + + } E 1 = ψ 2 + Uψ 2, inf ψ H 1 (R+ ), ψ L 2=1 d2 U dξ = ψ 1 2, 2 du U() =, dξ L2 (R + ). (..35) III.2 Résultats obtenus Les modèles (..31) et (..34) sont bien posés et admettent des solutions uniques. L étude du problème Schrödinger-Poisson sur un domaine borné est effectuée par F. Nier [84, 85, 86] en le reformulant en un problème de minimisation convexe. Le système limite (..35) quand à lui est posé sur [, [. Le premier résultat du Chapitre 5 concerne l étude de ce modèle (Théorème 5.1.1). Nous sommes amenés à étudier aussi un problème de minimisation posé sur un domaine non borné. Cette étude est réalisée grace au principe de concentration-compacité introduit par P. L. Lions [79]. Le deuxième résultat concerne la comparaison de différents systèmes présentés ci-dessus. Nous obtenons le théorème suivant. Theorem..1. Soit V ε, Ṽε et U les potentiels satisfaisant (..31), (..34) et (..35) respectivement. Les estimations suivantes sont vérifiées V ε Ṽε H 1 (,1) = O(e c ε 2 ),

39 REFERENCES 39 Ṽε 1 ε 2 U (. ε ) H 1 (,1) = O(e c ε ), où c est une constante générique strictement positive et indépendante de ε. III.3 Commentaires Le choix de condition au bord de type Newmann pour le potentiel au point z = 1 est justifié dans certaines situations physiques [5]. Les conditions de Dirichlet sont plus habituelles dans ce type de modèle. Dans ce cas, l analyse peut être effectuée mais comporte des complexités techniques liées au fait qu une autre couche limite apparaît au point z = 1 et que les valeurs propres admettent asymptotiquement une multiplicité double. Pour simplifier, nous avons donc imposé des conditions de type Newmann en z = 1. Il est plus naturel d utiliser une statistique de Fermi-Dirac dans la limite forte densité. Cette étude peut être généralisée au cas Fermi-Dirac avec quelques complications techniques (voir dernière section du Chapitre 5 pour plus de détails). Enfin, le problème multidimensionnel est beaucoup plus compliqué. La localisation de la couche limite dans ce cas peut dépendre de la géométrie de bord du domaine considéré. Ce type de problème est présent dans l étude de l équation de Schrödinger avec champ magnétique [21, 75]. References [1] T. Ando, A. B. Fowler, and F. Stern, Electronic properties of twodimensional electron systems, Rev. Mod. Phys., 54, 2 (1982), p [2] C. Bardos, F. Golse, and C. D. Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), pp [3] C. Bardos, F. Golse, and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys., 63 (1991), pp [4], The acoustic limit for the Boltzmann equation, Arch. Rational. Mech. Ana., 153 (2), pp [5] G. Bastard, Wave mechanics applied to semiconductor heterostructures, Les éditions de physiques, [6] N. Ben Abdallah, A hybrid kinetic-quantum model for stationary electron transport, J. Stat. Phys., 9 (1998), pp [7], On a multidimensional Schrödinger-Poisson scattering model for semiconductors, J. Math. Phys., 41 (2), pp

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43 REFERENCES 43 [49] R. J. Elliott, Theory of the Effect of Spin-Orbit Coupling on Magnetic Resonance in Some Semiconductors, Phys. Rev., 96 (1954), pp [5] C. Fermanian-Kammerer and P. Gérard, Mesures semi-classiques et croisement de modes, Séminaire Équations aux dérivées partielles (Polytechnique) (1999-2), Art. No. 17, 13 p. [51], Une formule de Landau-Zener pour un croisement non dégénéré et involutif de codimension 3, C. R. Math. Acad. Sci. Paris, 335 (22), pp [52], A Landau-Zener formula for non-degenerated involutive codimension 3 crossings, Ann. Henri Poincaré, 4 (23), pp [53] C. Fermanian-Kammerer and P. Gerard, A Landau-Zener formula for two-scaled Wigner measures, in Dispersive transport equations and multiscale models (Minneapolis, MN, 2), vol. 136 of IMA Vol. Math. Appl., Springer, New York, 24, pp [54] C. Fermanian-Kammerer and P. Gérard, Two-scale Wigner measures and the Landau-Zener formulas, in Multiscale methods in quantum mechanics, Trends Math., Birkhäuser Boston, Boston, MA, 24, pp [55] A. Fert, A. Friederich, and al, Giant Magnetoresistance of (1)Fe/(1)Cr Magnetic Superlattices, Phys. Rev. Lett., 61 (1988), p [56] G. Fishman and G. Lampel, Spin relaxation of photoelectrons in p-type gallium arsenide, Phys. Rev. B, 16 (1977), pp [57] E. Frenod and K. Hamdache, Homogenisation of transport kinetic equations with oscillating potentials, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), pp [58] E. Frénod, P.-A. Raviart, and E. Sonnendrücker, Two-scale expansion of a singularly perturbed convection equation, J. Math. Pures Appl. (9), 8 (21), pp [59] E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal., 18 (1998), pp [6] E. Frénod and E. Sonnendrücker, Long time behavior of the twodimensional Vlasov equation with a strong external magnetic field, Math. Models Methods Appl. Sci., 1 (2), pp [61] E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (21), pp (electronic). [62] E. Frénod and F. Watbled, The Vlasov equation with strong magnetic field and oscillating electric field as a model for isotop resonant separation, Electron. J. Differential Equations, (22, No. 6), p. 2 (electronic).

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45 REFERENCES 45 [77] W. W. Lee, Gyrokinetic approach in particle simulation, Physics of Fluids, 26 (1983), pp [78] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), pp [79] P. Lions, The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 1, Ann. Inst. Henri Poincaré, analyse non linéaire, 1 (1984), pp [8] P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), pp [81] R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Phys. Fluids, 24 (1981), pp [82] G. Lommer, F. Malcher, and U. Rössler, spin splitting in semiconductor heterostructures for B, Phys. Rev. Lett., 6 (1988), pp [83] B. Lucquin-Desreux, Diffusion of electrons by multicharged ions, Math. Models Methods Appl. Sci., 1 (2), pp [84] F. Nier, A stationary Schrödinger-Poisson System Arising from the modelling of electronic devices, Forum Math., 2 (199), pp [85], A variational formulation of Schrödinger-Poisson Systems in Dimension d 3, Comm. Partial Differential Equations, 18 (1993), pp [86], Schrödinger-Poisson Systems in Dimension d 3 :The Whole-space case, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), pp [87] T. G. Northrop, The guiding center approximation to charged particle motion, Annals of Physics, 15 (1961), pp [88] Y. V. Pershin, Accumulation of electron spin polarization at semiconductor interfaces, Phys. Rev. B, 68 (23), p [89] Y. V. Pershin and V. Privman, Focusing of spin polarization in semiconductors by inhomogeneous doping, Phys. Rev. Lett., 9 (23), p [9] P. Pietra and N. Vauchelet, Modeling and simulation of the diffusive transport in a nanoscale double-gate mosfet, submitted. [91] O. Pinaud, Adiabatic approximation of the Schrödinger-Poisson system with a partial confinement : the stationnary case, J. Math. Phys., 45 (24), p [92] E. Polizzi, Modélisation et simulation numériques du transport quantique balistique dans les nanostructures semi-conductrices, PhD. Thesis, INSA, Toulouse, France, (21).

46 46 REFERENCES [93] E. Polizzi and N. Ben Abdallah, Self-consistent three dimensional model for quantum ballistic transport in open systems, Phys. Rev B., 66 (22), pp [94], subband decomposition approach for the simulation of quantum electron transport in nanostructures, J. Comp. Phys., 22 (25), pp [95] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation : analysis of boundary layers, Asymptotic Anal., 4 (1991), pp [96] L. Saint-Raymond, The gyrokinetic approximation for the Vlasov-Poisson system, Math. Models Methods Appl. Sci., 1 (2), pp [97] L. Saint-Raymond, Control of large velocities in the two-dimensional gyrokinetic approximation, J. Math. Pures Appl. (9), 81 (22), pp [98] N. Vauchelet, Modélisation mathématique du transport diffusif de charges partiellement quantiques, Thesis, Université Paul Sabatier-Toulouse III, (26). [99] C. Villani, Limite hydrodynamiques de l équation de Boltzmann, Séminaire Bourbaki, 53ème année (2 21), p. n 893. [1], A review of mathematical topics in collisional kinetic theory, in Handbook of mathematical fluid dynamics 1, North-Holland Amsterdam, 22. [11] Z. G. Yu and M. E. Flatté, Electric-field dependent spin diffusion and spin injection into semiconductors, Phys. Rev. B, 66 (22), p [12] Z. G. Yu and M. E. Flatté, Spin diffusion and injection in semiconductor structures : electric field effects, Phys. Rev. B, 66 (22), p [13] I. Zutic, J. Fabian, and S. D. Sarma, Spintronics : Fundamentals and applications, Reviews of Modern Physics, 76 (24), p. 323.

47 Part I Transport models for semiconductor spintronics 47

48

49 Chapter 1 Semiclassical analysis of the Schrödinger equation with spin-orbit hamiltonian Abstract The aim of this first chapter is the derivation of kinetic type models for semiconductor spintronics. In the first part, a semi-classical analysis of the Schrödinger equation with general spin-orbit hamiltonian is performed. Two situations are discussed. First, if the spin-orbit hamiltonian (H SO ) is considered as a perturbative term with order ε (the scaled planck constant) of the total energy hamiltonian (H ), then the semi-classical limit (ε ) leads to the so called spinor kinetic equation. However if H SO = O(H ), a two component kinetic model is obtained at the limit. The second part of this chapter is dedicated to the study of the semi classical limit of a partially confined electron gas in one direction of the space (z). A partial semi-classical limit is applied following the work of N. Ben Abdallah and F. Méhats [1]. The limit model in this case couples an infinity quasi-static Schrödinger equations describing the energy levels in the confinement direction and an infinity Vlasov equations in the transport directions with Rashba spin-orbit coupling. Moreover, an explicit expression relying the order of the Rashba effect and the confinement potential is derived. 49

50 5 CHAPTER 1. SEMICLASSICAL ANALYSIS 1.1 Introduction The study of spin related phenomenae in semi-conductor devices attracts a huge attention by the physical community. The main reason is the possibility to control the spin vector direction due to the existence of mechanisms acting on the spin dynamics. The must important mechanism existing is the spin-orbit coupling. This kind of spin-dependent interactions results generally from a lack of symmetry in the structure. It manifests itself by the appearance of an effective magnetic field which makes precess the spin vector during the free paths of the particles. There are two main types of spin-orbit interactions in semiconductor heterostructures. The Dresselhaus spin-orbit interactions [5] results from asymmetry present in certain crystal lattices like zing blende structures. The Rashba spin-orbit interactions arises due to the asymmetry of quantum wells formed at the interface of semiconductor heterostructures [3]. The spin-orbit hamiltonian takes the following general form [24] H SO = α Ω(t, x, k) σ (1.1.1) where σ is the vector of the Pauli spin matrices given by (A.2.1), (t, x, k) are respectively the time, the position and the wave vector of the electron (k i x ), α is the strength of the spin-orbit coupling and Ω represents the effective field or the precession vector. Let us give some examples. In the Elliot-Yafet mechanism of relaxation [24], the spin-orbit hamiltonian is given by 2 H SO = i 4m 2 c ( xv 2 x ) σ. (1.1.2) The Rashba spin-orbit hamiltonian writes as [3] H R = αi (σ 1 x2 σ 2 x1 ), and the Dresselhauss spin-orbit interactions can be represented by the following hamiltonian [5] H D = αi (σ 1 x1 σ 2 x2 ). In this chapter, different kinetic models describing the spin transport in semiconductor spintronic devices will be rigorously derived. A semi-classical analysis of the Schrödinger equation with spin-orbit hamiltonian using the theory of Wigner transform is performed. It leads, according to the order of the spin-orbit coupling with respect to the scaled Planck constant, either to a spinor kinetic equation with 2 2 hermitian matrix value distribution function or to a two-component kinetic model. This is the subject of the next section. The third section is concerned with the derivation of subband kinetic/quantum model for partially confined systems.

51 1.2. SCHRÖDINGER EQUATION WITH GENERAL SPIN-ORBIT HAMILTONIAN Schrödinger equation with general spin-orbit Hamiltonian This part is dedicated to the study of the semi classical limit of the Schrödinger equation with spin-orbit hamiltonian. The starting equation will be the following scaled linear Schrödinger equation : iε t Ψ ε = ε2 2 xψ ε + V ε Ψ ε + αω W ε (t, x, iε x ) σψ ε (1.2.1) subject to the following initial condition Ψ ε (t = ) = Ψ ε I. (1.2.2) Here, Ω W ε (t, x, iε x ) σ is the Weyl operator associated to the complex 2 2 matrix valued symbol Ω ε (t, x, ξ) σ on R 3 x R 3 ξ defined by Ω W ε (t, x, iε x ) σ(ψ) = 1 ( Ω (2π) 3 ε t, x + y ), εξ σ(ψ(y))e i(x y).ξ dξdy. R 3 ξ R 2 3 y (1.2.3) This operator represents the spin-orbit hamiltonian with Ω ε is the effective field and α is the spin-orbit coupling order. We recall that the wave function Ψ ε (t, x) = (ψ ε (t, x), ψ ε (t, x)) is a function of the time t R + and the position x R 3 with C 2 vector value. The potential V ε is given and regular (satisfying Assumption 1.2.4). The parameter ε is a scaled Planck constant intended to go to zero. To perform rigourously this semi classical limit, we will apply some interesting convergence results using Wigner transform techniques. A review of these techniques can be found in [13]. The Wigner transform is a powerful tool introduced initially by Wigner in [23]. Many important convergence results for semi classical limit have been then obtained using this transform. We cite for instance the works of Gérard [12], Lions and Paul [15] and Markowich and Mauser [16]. The order of the spin-orbit coupling with respect to ε will obviously play an important role in this semi classical limit. We treat here two cases : α = O(ε) and α = O(1). In the first case (α = O(ε)), the spin-orbit coupling ( is considered ) as a ξ perturbation part of the total energy symbol P ε 2 (t, x, ξ) = 2 + V ε (t, x) I 2 and the semi classical limit leads to the spinor kinetic equation. This result is cited in Theorem However, when α = O(1), a splitting between the two energy levels of the spin-up and spin-down species will occur with energy gap of order Ω where Ω denotes the limit of Ω ε when ε goes to zero. The semi-classical limit leads to a two-component kinetic model in this case. This is the subject of the second theorem of this section, Theorem

52 52 CHAPTER 1. SEMICLASSICAL ANALYSIS Theorem (case α = O(ε)). Under Assumptions 1.2.4, 1.2.5, and 1.2.1, we have the following results. 1. The Wigner transform W ε (Ψ ε I, Ψε I ), defined by (1.2.12), converges in S (R 3 x R 3 ξ, M 2(C)) to W I M b (R 3 x R 3 ξ, M 2(C)). 2. The Wigner transform W ε (Ψ ε, Ψ ε ) of the solution of (1.2.1)-(1.2.2) converges in L (R +, S (R 3 x R 3 ξ, M 2(C))) weak topology to W C (R +, M b (R 6, H + 2 (C))). Moreover, if α = O(ε) then, W satisfies the spinor Boltzmann equation : t W + ξ x W x V ξ W = i[w, Ω σ] (1.2.4) W (, x, ξ) = W I (1.2.5) where V and Ω denote respectively the limits of V ε and Ω ε as ε. 3. For every T >, the macroscopic quantities N ε, J ε given by Definition converge in L ((, T ); M b (R 3 x, M 2 (C))) weak topology to : and N(t, x) = W (t, x, ξ)dξ, R 3 J(t, x) = vw (t, x, ξ)dξ. R 3 In this theorem as in the sequel, if E is a Banach space, S (R 3, E) denotes the E-valued tempered distributions and M b (R 3, E) is the space of E-valued bounded measures. Moreover, M 2 (C) denotes the space of 2 2 complex matrices and H + 2 (C) is the space of hermitian positive definite matrices. The second Theorem is concerned with the case α = O(1). Theorem (case α = O(1)). Under the same assumptions as for Theorem and if α = O(1) with respect to ε, then the spin part of the Wigner measure associated to (Ψ ε ) ε, W, given by the above theorem is parallel to Ω outside the set { E = (t, x, ξ) R + R 6 / Ω(t, } x, ξ) =. (1.2.6) In other words, we have W (t, x, ξ) = 1 2 w Ω c(t, x, ξ)i 2 +w s (t, x, ξ) for all (t, x, ξ) Ω σ R + R 6 \ E with E is the closure of E. In addition, the eigenvalues of W given by w = w c 2 + w s and w = w c 2 w s satisfy the following two-kinetic model t w + ξ λ x w x λ ξ w = on (R + R 6 x,ξ ) \ E (1.2.7) t w + ξ λ x w x λ ξ w = on (R + R 6 x,ξ ) \ E.

53 1.2. SCHRÖDINGER EQUATION WITH GENERAL SPIN-ORBIT HAMILTONIAN 53 The up and down total energies, λ and λ, are respectively given by λ (t, x, ξ) = ξ V + Ω, λ (t, x, ξ) = ξ V Ω. (1.2.8) Remark Outside E = {(t, x, ξ) R + R 6 / Ω(t, x, ξ) = }, the eigenvalues λ and λ are distincts and of constant multiplicity 1. The distribution functions w and w propagate along the hamiltonian curves (or the characteristics curves) associated with λ and λ outside E. However, when the classical trajectories reach E, a crossing of the two energy levels occurs and some energy transfer between the two modes is expected to happen above the crossing. Many works have been devoted to describe the semi-classical evolution through crossings and to quantify the energy transfer in terms of two-scale Wigner measures and Landau-Zener formula. We refer the reader to the works of Patrick Gérard and Clotilde Fermanian-Kammerer [6, 7, 8, 9, 1] and references therein Analysis of the Schrödinger equation with spin-orbit term This subsection is concerned with the analysis of (1.2.1)-(1.2.2). Some assumptions on the potential V ε, the initial data Ψ ε I and the effective field of the spin-orbit coupling Ω ε are needed. Assumption We assume that (ε, t, x) V ε (t, x) is a real nonnegative function belonging to C ([, 1] ε, C 1 W 1, (R + R 3 )). Assumption We assume that the initial data (Ψ ε I ) ε belongs to H 1 (R 3, C 2 ) and satisfies the following energy estimate E ε (Ψ ε I) := ( Ψ ε I 2 + ε 2 x Ψ ε I 2 2)dx C, R 3 for some constant C > independent on ε. In this text,. and. 2 denote respectively the Euclidian vector and matrix norms. Assumption We suppose that Ω ε (t, x, ξ) belongs to C ([, 1] ε, C 1 (R + R 6 x,ξ ))3. In addition, there exists m > such that for all α, β N with α+β 1, there exists C α,β (t) a positive continuous function independent of ε such that l, k {1, 2, 3} α+β Ω x α ε C α,β (t)(1 + ξ ) m β. (1.2.9) k ξβ l

54 54 CHAPTER 1. SEMICLASSICAL ANALYSIS Definition Let Ψ ε associated particle and current densities are defined by C (R +, H 1 (R 3 x, C 2 )) be the solution of (1.2.1). The and N ε (t, x) = Ψ ε (t, x) Ψ ε (t, x) (1.2.1) J ε (t, x) = ε 2i [ xψ ε (t, x) Ψ ε (t, x) Ψ ε (t, x) x Ψ ε (t, x)]. (1.2.11) Definition Let φ 1, φ 2 S (R 3 x, C 2 )(the space of tempered distributions), the Wigner matrix associated to φ 1 and φ 2 is the M 2 (C)-valued distribution on R 3 x R 3 ξ defined by W ε (φ 1, φ 2 )(x, ξ) = (2π) R 3 e iη ξ φ 1 (x ε η 2 ) φ 2(x + ε η )dη. (1.2.12) 2 3 We refer the reader to [13, 15] for a detailed study of this transformation and its application to the semi classical analysis. The first proposition ensures the existence of solution of (1.2.1)- (1.2.2) and gives energy estimate. Proposition Under Assumptions 1.2.4, and for any ε [, 1], (1.2.1)- (1.2.2) admits a unique weak solution Ψ ε C (R +, H 1 (R 3, C 2 )) C 1 (R +, H 1 (R 3, C 2 )) (1.2.13) and the sequence (Ψ ε ) ε is bounded in C (R +, L 2 (R 3, C 2 )). Proof. Since the Pauli matrices are hermitian, it is simple to verify that the hamiltonian H ε = ( ε2 2 x +V )I 2 +αω W (t, x, iε ) σ (with I 2 is the 2 2 identity matrix) is a self adjoint operator on its domain H 2 (R 3, C 2 ) in L 2 (R 3, C 2 ). Then, applying the Stone s Theorem (see [22, 2]), equation (1.2.1) admits a unique weak solution satisfying (1.2.13). Moreover, taking the scalar product of (1.2.1) with Ψ ε in C 2, integrating with respect to x and taking the imaginary part, one gets t ( Ψ ε 2 dx) = R which gives the mass conservation 3 Ψ ε 2 dx = R 3 Ψ ε I 2 dx. R 3 (1.2.14) Assumption The energy estimate is propagated with the weak solutions (Ψ ε ) ε of (1.2.1)-(1.2.2). This means that there exists a continuous function C(t) independent of ε such that t, E ε (Ψ ε ) = ( Ψ ε 2 + ε 2 x Ψ ε 2 2)dx C(t). R 3 (1.2.15)

55 1.2. SCHRÖDINGER EQUATION WITH GENERAL SPIN-ORBIT HAMILTONIAN 55 Notice that Assumption is verified in practice when taking Rashba, Dresselhauss, or Elliot-Yafet spin-orbit hamiltonian. Indeed, taking for instance iε t Ψ ε = ε2 2 xψ ε + V ε Ψ ε + iε 2 ( σ x V ε ). x Ψ ε multiplying by t Ψ ε, integrating with respect to x and taking the real part, one finds ε 2 4 t x Ψ ε 2 2dx+ 1 ( ) V ε t ( Ψ ε 2 )dx = ε 2 I ( σ x V ε ). x Ψ ε. t Ψ R 2 ε dx 3 R 3 R 3 (1.2.16) where I(z) denotes the imaginary part of z C. In addition, by an integration by part ( t ) I ( σ x V ε ) x Ψ ε s Ψ ε dsdx (1.2.17) R 3 = 1 t [ ( σ x V ε ) x Ψ ε s Ψ 2i ε ( σ x V ε ) x Ψ ε s Ψ ε] dsdx R 3 = 1 t ] [ s (( σ x V 2i ε ) x Ψ ε ) Ψ ε Ψ ε ( σ x V ε ) x s Ψ ε dsdx R 3 1 ( σ x V 2i ε ) x Ψ ε (t, x) Ψ ε (t, x)dx + 1 ( σ x V R 3 2i ε (, x)) x Ψ ε I Ψε Idx R 3 = 1 t ( σ x s V 2i ε ) x Ψ ε Ψ ε dsdx 1 ( σ x V R 3 2i ε ) x Ψ ε (t, x) Ψ ε (t, x)dx R ( σ x V 2i ε (, x)) x Ψ ε I Ψε Idx. R 3 Moreover, it is simple to show that for any regular function U we have ( ) 1 i ( σ x U) x Ψ 2 ε Ψ ε dx sup ( x U ) x Ψ ε 2 2 2dx R 3 R 3 R 3 ( R3 Ψ ε 2 ) 1 2. Applying this inequality to all the terms of the right hand side of (1.2.17), one finds thanks to Assumptions and and Young s formula ( t ) ε 2 I ( σ x V ε ) x Ψ ε s Ψ ε dsdx R 3 t C 1 (t) + ε2 8 ( x Ψ ε 2 2dx + x Ψ ε 2 2dsdx), (1.2.18) R 3 R 3 with a continuous function C 1 (t) independent of ε. Finally, integrating (1.2.16) with respect to t and with (1.2.18), we obtain ε 2 R 3 x Ψ ε 2 2dx C 2 (t) + ε 2 t R 3 x Ψ ε 2 2dsdx. Applying the Gronwall s Lemma and with the mass conservation (1.2.14), one deduces (1.2.15).

56 56 CHAPTER 1. SEMICLASSICAL ANALYSIS Semiclassical limit In this subsection, a brief proof of the two above theorems will be presented. We use a main theorem in the semi-classical analysis due to Gérard, Markovich and al in [13] to passing to the limit. More precisely we will use Theorem 6.1 of [13]. Proof of Theorem and Theorem The first point of Theorem ( results ) from the boundedness of Ψ ε I in L2 (R 3, C 2 ) (see [13]). Let P ε (t, x, ξ) = ξ V ε I 2, then equation (1.2.1) rewrites as ε t Ψ ε + i ( Pε W (t, x, iε x ) + αω W ε (t, x, iε x ) σ ) (Ψ ε ) =. (1.2.19) We begin by the first case : α = O(ε) and we take α = ε for simplicity. By taking a Hilbert expansions of V ε and Ω ε with respect to ε V ε = V + εv 1 + o(ε), Ωε = Ω + ε Ω 1 + o(ε), we have P ε + εω ε σ = P (t, x, ξ) + ε(v 1 I 2 + Ω σ) + o(ε), ( ) ξ 2 with P (t, x, ξ) = + V (t, x) I 2. Let λ(t, x, ξ) = ξ 2 + V (t, x) be the unique 2 2 eigenvalue of P (t, x, ξ) of multiplicity 2, and applying Theorem 6.1 of [13] then, W ε (Ψ ε, Ψ ε ) converges weak in L (R +, S (R 3 x R 3 v, M 2 (C))) to W C (R +, M b (R 6, H 2 + (C))). In addition, W satisfies t W + { λ, W } = [W, F λ ] on (R + R 6 x,ξ) \ E, (1.2.2) with F λ = [Π λ, {λ, Π λ }] + iπ λ (V 1 I 2 + Ω σ)π λ, Π λ is the orthogonal projection on the eigenspace associated to λ, and { } denotes the Poisson bracket : { } λ, W = ξ λ x W x λ ξ W. The set E represents a closed subset of R + R 3 x R 3 ξ such that, for every (t, x, ξ) / E, the eigenvalues of P can be ordered as follows, see [13], λ 1 (t, x, ξ) <... < λ d (t, x, ξ) where, for 1 q d, the multiplicity of λ q (t, x, ξ) does not depend on (t, x, ξ). Here, we have one eigenvalue λ(t, x, ξ) (d = 1) of multiplicity two for every (t, x, ξ) R + R 6. We choose then E =. Moreover, we have Π λ = I 2 then, F λ = i(v 1 I 2 + Ω σ) and equation (1.2.2) yields (1.2.4). For the convergence of the macroscopic quantities, we refer the reader also to [13].

57 1.2. SCHRÖDINGER EQUATION WITH GENERAL SPIN-ORBIT HAMILTONIAN 57 We assume now that α = O(1) and we take α = 1, then the principal order (with respect to ε) of the symbol of the pseudo-differential operator in equation (1.2.19) multiplied by i is given by P (t, x, ξ)+ Ω(t, x, ξ) σ. It admits two simple eigenvalues λ and λ given by λ = ξ V (t, x) + Ω(x, ξ), λ = ξ V (t, x) Ω(x, ξ). Denoting by Π and Π the orthogonal projections on the eigenspaces associated to λ and λ respectively, one can easily verify that Π = 1 2 ( I 2 + Ω σ Ω ), and Π = 1 2 ( I 2 Ω σ Ω ) on (R + R 6 ) \ E where E is given by (1.2.6). From Theorem 6.1 of [13], the weak limit of W ε (Ψ ε, Ψ ε ), W, is given by W = W + W. The up and down functions W and W are a two continuously t-dependent positive matrix valued measures on R 6 x,ξ satisfying t W ( ) + {λ ( ), W ( )} = [W ( ), F ( ) ] on (R + R 6 x,ξ) \ E, (1.2.21) with F ( ) = [Π ( ), {λ ( ), Π ( ) }] + iπ ( ) (V 1 I 2 + Ω 1 σ)π ( ). Moreover, we have W ( ) = Π ( )W Π ( ) and if we pose W = w c 2 I 2 + w s σ, then a straightforward calculation gives W = ( w c 2 + w s Ω ) ( Ω Π and W w c = 2 w s Ω ) Ω Π on (R + R 6 x,ξ) \ E. (1.2.22) We deduce that ( W = 1 w c w s Ω ) ( Ω Ω I 2 + ) σ Ω = w c 2 I 2 + w s Ω Ω 2 Ω σ on (R + R 6 x,ξ) \ E. ( w c 2 w s Ω ) ( Ω Ω I 2 ) σ Ω This implies that the spin part of W, w s is parallel to Ω. Let w s (t, x, ξ) = w s Ω Ω and let w = w c 2 + w s and w = w c 2 w s be the eigenvalues of W, we have, from (1.2.22), W = w Π and W = w Π. Equation (1.2.21) yields (1.2.7). Indeed, we have t W ( ) + {λ ( ), W ( )} = ( t w ( ) + {λ ( ), w ( ) })Π ( ) + w ( ) {λ ( ), Π ( ) }.

58 58 CHAPTER 1. SEMICLASSICAL ANALYSIS In addition, using the identity [ a σ, b σ] = 2i( a b) σ for any two vectors a and b, one has [W, F ] = [w Π, [Π, {λ, Π }]] + [w Π, iπ (V 1 I 2 + Ω 1 σ)π ] [ [ { } ]] = w Ω σ Π, 4 Ω, Ω λ, Ω σ + [ ( { }) ] = iw Ω Π, 2 Ω Ω λ, Ω σ ( { }) = w Ω Ω 2 Ω Ω Ω λ, Ω σ ({ } = w Ω Ω λ, 2 Ω. ) Ω σ Ω Ω + w Ω. { } Ω Ω 2 Ω λ, 2 Ω σ { } = w Ω λ, 2 Ω σ = w {λ, Π }. Similarly, [W, F ] = w {λ, Π } and we conclude that ( t w ( ) + {λ ( ), w ( ) })Π ( ) + w ( ) {λ ( ), Π ( ) } = w ( ) {λ ( ), Π ( ) } which gives system (1.2.7). 1.3 Semi-classical limit of the Schrödinger equation with Spin-Orbit term and partially confining potential Introduction and main result The starting equation in this section is the following scaled Schrödinger equation iε t Ψ ε = ε2 2 xψ ε zψ ε + V ε Ψ ε + εω W ε (t, x, z, i ε ) σ(ψ ε ) Ψ ε (, x, z) = Ψ ε I (x, z) (1.3.1) where (x, z) R 2 [, 1], Ψ ε = Ψ ε (t, x, z) and ε = (ε x, z ). Let us denote the spatial domain by D = R 2 (, 1). The position variables in the longitudinal directions are denoted by x R 2 and z is used for the transversal direction (or confinement one). For any ξ R 3, we

59 1.3. SEMICLASSICAL LIMIT AND PARTIALLY CONFINING POTENTIAL 59 will write ξ = (ξ x, ξ z ) where ξ x R 2 and ξ z R are respectively the longitudinal and transversal parts of ξ. The Weyl operator Ω W ε (t, x, z, i ε ) σ represents in this section the following operator Ω W ε (t, x, z, i ε ) σ(ψ) = 1 ( Ω (2π) 2 ε t, x + y, z, εξ x, i z ) σ(ψ(y, z))e i(x y).ξx dξ x dy. R 2 R ξx 2 2 y Here, for every (t, x, z, ξ x ) R + D R 2 ξ x, Ω(t, x, z, ξ x, i z ) is a vector valued differential operator which is the image of i z by some regular vector valued function Ω(t, x, z, ξ x,.). The confinement in the z direction is modeled through the following boundary conditions Ψ ε (t, x, z = ) = Ψ ε (t, x, z = 1) =. (1.3.2) Thanks to this assumption, the transverse Hamiltonian ( z +V ε )I 2 has a discrete spectrum and a complete set of eigenfunctions. Let χ ε p(t, x,.) and ɛ ε p(t, x) be the eigenfunctions and eigenvalues of z +V ε with homogenous boundary conditions, they satisfy zχ ε p + V ε χ ε p = ɛ ε pχ ε p χ ε p(t, x,.) H 1 (, 1), 1 χ ε pχ ε q = δ pq. (1.3.3) For any (t, x) R + R 2, (χ ε p(t, x,.)) p 1 is an orthonormal basis of L 2 (, 1), (ɛ ε p) p 1 are simple as eigenvalues of z +V ε. The eigenvalues of the transverse Hamiltonian ( z + V ε )I 2 are then ɛ ε p and are of multiplicity two. Let us denote by H ε p(t, x) the p-th eigenspace which is given by H ε p(t, x) = span {( χ ε p ), ( χ ε p )} (L 2 (, 1)) 2 and by Π ε p(t, x) the orthogonal projector on H ε p(t, x). For ε =, we shall use the notation Π p and ɛ p instead of Π p and ɛ p. As in the previous section we take the following assumptions Assumption For any T >, (ε, t, x, z) V ε (t, x, z) is a real nonnegative function belonging to C ([, 1] ε, C 1 W 1, ([, T ] R 2, L (, 1))). Assumption The initial data Ψ ε I belongs to H1 (D, C 2 ) and there exists C > independent of ε such that ( Ψ ε I 2 + ε 2 x Ψ ε I z Ψ ε I 2 )dxdz C. D

60 6 CHAPTER 1. SEMICLASSICAL ANALYSIS Assumption The effective field Ω ε (t, x, z, ξ x, ξ z ) belongs to C ([, 1] ε, C 1 (R + D R 3 ξ ))3 such that (1.2.9) holds. In addition, we assume that Ω ε satisfies the following expansion Ω ε = Ω(t, x, z, ξ x ) + εω 1 (t, x, z, ξ x, ξ z ) + o(ε) (1.3.4) with Ω C 1 (R + D R 2 ξ x ) and Ω 1 C 1 (R + D R 3 ξ x,ξ z ). Definition Let H = L 2 ((, 1), C 2 ). This is an Hilbert space equipped with the L 2 norm. The unknown Ψ ε of (1.3.1) is an H-valued function on R + R 2. In the sequel, H H will denote the space of bilinear forms on H H. It can be identified to L 2 ((, 1) 2, M 2 (C)) : the set of linear combination of simple tensors of the form (Φ 1 Φ 2 )(z, z ) for any Φ 1, Φ 2 H and where (Φ 1 Φ 2 )(Ψ 1, Ψ 2 ) = = Ψ 1, Ψ 2 H. 1 1 Φ 1 (z) Φ 2 (z ) : Ψ 1 (z) Ψ 2 (z )dzdz ( 1 ) ( ) 1 Φ 1 (z) Ψ 1 (z)dz Φ 2 (z ) Ψ 2 (z )dz Definition Let Φ 1, Φ 2 be two H-valued distribution belonging to S (R 2 x, H). For any ε >, the partial Wigner transform is given by W ε (Φ 1, Φ 2 )(x, ξ x, z, z) = (2π) R 2 e iη ξx Φ 1 (x ε η 2, z) Φ 2(x + ε η 2, z )dη. (1.3.5) 2 This defines a continuous sesquilinear mapping from S (R 2 x, H) S (R 2 x, H) to S (R 2 x R 2 ξ x, H H). Under Assumptions 1.3.1, 1.3.2, equation (1.3.1) admits a unique weak solution Ψ ε C (R +, H(D, 1 C 2 )) C 1 (R +, H 1 (D, C 2 )) satisfying the mass conservation (see the previous section). We assume moreover that the energy estimate is propagated t >, ( Ψ ε 2 + ε 2 x Ψ ε z Ψ ε I 2 )dxdz C(t) (1.3.6) D where C(t) is a continuous function of t independent of ε. Definition Let Ψ ε be the solution of (1.3.1). The associated macroscopic quantities are now given by N ε (t, x, z, z ) = Ψ ε (t, x, z) Ψ ε (t, x, z ) (1.3.7) and J ε (t, x, z, z ) = ε 2i [ xψ ε (t, x, z) Ψ ε (t, x, z ) Ψ ε (t, x, z) x Ψ ε (t, x, z )]. (1.3.8)

61 1.3. SEMICLASSICAL LIMIT AND PARTIALLY CONFINING POTENTIAL 61 We are now read to give the main result of this section. This is the subject of the next Theorem. Theorem Assume that Assumptions 1.3.1, and hold. Then, up to extraction of subsequences, we have the following results at the limit ε. 1. For any p N, W ε (Π ε p(, x)ψ ε I, Πε p(, x)ψ ε I ) converges in S (R 4 x,ξ x, H H) to F p,i (x, ξ x )χ p (, x, z)χ p (, x, z ) M b (R 4 x,ξ x, H H). Moreover, F p,i (x, ξ x ) H 2 + (C) for all (x, ξ x ) R The Wigner transform W ε (Ψ ε, Ψ ε ) of the solution of (1.3.1) converges in L (R +, S (R 4 x,ξ x, H H)) weak to W = p F p (t, x, ξ x )χ p (t, x, z)χ p (t, x, z ) C (R +, M b (R 4 x,ξ x, H H)) with F p (t, x, ξ x ) H 2 + (C). It solves t F p + ξ x x F p x ɛ p ξx F p = i[f p, Ω p σ], F p (, x, ξ x ) = F p,i (x, ξ x ), (1.3.9) (1.3.1) where the effective field on the p-th subband, Ω p, is given by Ω p (t, x, ξ x ) = 1 Ω(t, x, z, ξ x ) χ p (t, x, z) 2 dz. (1.3.11) 3. For every T >, the particle and current densities N ε and J ε given by (1.3.7)- (1.3.8) converge in L ((, T ), M b (R 2 x, H H)) weak to N(t, x, z, z ) = ( ) F p (t, x, ξ x )dξ x χ p (t, x, z)χ p (t, x, z ) p 1 R 2 J(t, x, z, z ) = ( ) vf p (t, x, ξ x )dξ x χ p (t, x, z)χ p (t, x, z ). p 1 R Application : subband model with Rashba spin-orbit effect If we consider the general spin-orbit hamiltonian given by (1.1.2), the Schrödinger equation with physical dimensional variables writes 2 i t Ψ ε = 2 2m x,zψ ε + V ε Ψ ε + i 4m 2 c ( σ x,zv ε ) 2 x,z Ψ ε (1.3.12) with is the Planck constant, m is the effective mass of an electron and c denotes the speed of light. Let respectively, L and l be the longitudinal and transversal length

62 62 CHAPTER 1. SEMICLASSICAL ANALYSIS scales. Here, we are interested in situations where the length scale in the confinement direction z, l, is much smaller than the length scale in the non confinement one. In other words, let ε be the ratio between the two length scales l and L, we assume that ε := l L << 1. This leads, up to rescaling of equation (1.3.12), to iε t Ψ ε = ε2 2 xψ ε zψ ε + V ε Ψ ε + iβ( σ ε V ε ) ε Ψ ε (1.3.13) 2 where ε = (ε x, z ) and β =. If V and t denote the potential and time 4m 2 c 2 l2 scales, to obtain (1.3.13) we have taken the following hypotheses V = h2 ml 2 and V t = 1 ε. The constant β is a dimensionless quantity. We shall assume that β is small and is of order ε and we take β = ε for simplicity. A straightforward computation gives ( σ ε V ε ) ε ψ ε = z V ε (σ 2 (ε x ) σ 1 (ε y ))Ψ ε +εσ 3 ( x V ε (ε y ) y V ε (ε x ))Ψ ε +ε(σ 1 y V ε σ 2 x V ε ) z Ψ ε where (x, y) are the components of x R 2. This implies that i( σ ε V ε ) ε = Ω W ε (t, x, z, i ε ) σ with Ω ε = Ω + εω 1 + o(ε) such that Ω(t, x, z, ξ) = z V ( ξx, 2 ξx, 1 ), Ω1 (t, x, z, ξ) = ( y V ξ z, x V ξ z, x V ξx 2 y V ξx) 1 and where V is the limit of V ε when ε goes to zero and ξ = (ξ 1 x, ξ 2 x, ξ z ). Applying Theorem 1.3.7, one finds the following Boltzmann equation in the p-th subband t F p + ξ x x F p x ɛ p ξx F p = iα p (t, x)[σ 2 ξ 1 x σ 1 ξ 2 x, F p ] where α p (t, x) = 1 z V (t, x, z) χ p (t, x, z) 2 dz. (1.3.14) Remark The spin-orbit effect obtained in this limit is nothing but the Rashba effect used generally in the literature [3, 24] to describe the spin-orbit interactions in bidimensional gases. Moreover, an explicit relation between the confinement potential and the strength of the Rashba spin-orbit coupling is obtained, relation (1.3.14).

63 1.3. SEMICLASSICAL LIMIT AND PARTIALLY CONFINING POTENTIAL Proof of Theorem Theorem can be proved by following the work of N. Ben Abdallah and F. Mehats [1]. We present here the ideas of the proof and we refer to [1] for details. For the general properties of the Wigner transform (1.3.5) we refer the reader to [1, 13, 15]. Let us summarize some of these properties. We introduce the following space of H H-valued test functions : A = {φ Cc (R 2 x R 2 ξ x, H H) / (F ξx φ)(x, η) L 1 (R 2 η, Cc (R 2 x, H H))} (1.3.15) with F ξx φ be the Fourier transform with respect to ξ x. This space equipped with F ξx φ L 1 (R 2 η,(cc (R 2 x,h H)) is a separable Banach space. Proposition Let (Ψ ε ) be a bounded family of L 2 (R 2 x, H) functions. Then, the sequence W ε (Ψ ε, Ψ ε ) is uniformly bounded in A. Namely, we have W ε (Ψ ε, Ψ ε ) A Ψ ε L 2 (R 2 x,h). (1.3.16) There exists a bounded measure with 2 2 hermitian and non negative matrix value, W, such that, up to extraction of subsequence, W ε (Ψ ε, Ψ ε ) converges to W in A weak ; W is called the Wigner measure associated to this subsequence of (Ψ ε ). Moreover, assume that (Ψ ε ) is ε oscillatory : ε 2 R 2 x Ψ ε 2 2dx C (1.3.17) with C > independent of ε and the following compactness property holds : lim n + lim sup (Id P ε n)ψ ε L 2 (R 2,H) =. (1.3.18) ε We denote by Id the identity function on H and (P ε n) n N is a sequence of functions in C ([, 1] ε R 2, Com(H)) such that lim P n = Id in the L(H) weak topology, n locally uniformly with respect to x. Then, N ε W (., ξ x )dξ x in M b (R 3, H H) weak, (1.3.19) R 3 J ε ξ x W (., ξ x )dξ x R 3 in M b (R 3, H H) weak, (1.3.2) where the quantities N ε and J ε are given by Definition

64 64 CHAPTER 1. SEMICLASSICAL ANALYSIS Lemma Let T > be fixed and let (φ ε 1(t, x)) and (φ ε 2(t, x)) be two families bounded in L ((, T ), L 2 (R 2 x, H)). Consider an L(H)-valued function (ε, t, x, ξ x ) P ε (t, x, ξ x ) belonging to C ([, 1], C 1 (R + R 4 x,ξ x, L(H))) and satisfying, for some M, α N 3 N 3 / α 1 : α x,ξ x P ε (x, ξ x ) L(H) C α (1 + ξ x ) M with C α > independent of ε. Then, we have W ε (P W ε (t, x, iε x )φ ε 1, φ ε 2) = P ε W ε (φ ε 1, φ ε 2) + ε 2i {P ε, W ε (φ ε 1, φ ε 2)} + εr ε 1 (1.3.21) W ε (φ ε 1, Pε W (t, x, iε x )φ ε 2) = W ε (φ ε 1, φ ε 2)Pε + ε 2i {W ε (φ ε 1, φ ε 2), Pε } + εr2 ε (1.3.22) where r1 ε and r2 ε in S (R 4, H H) weak uniformly with respect to t (, T ). Here, {p, q} denotes the Poisson bracket of two functions p(x, ξ) and q(x, ξ), given by {p, q}(x, ξ) = ξ p(x, ξ). x q(x, ξ) x p(x, ξ). ξ q(x, ξ). (1.3.23) Before going on, one needs the following lemma which summarizes the regularities of the eigenelements of the transversal Schrödinger operator. Lemma Let (ɛ ε p, χ ε p) ε satisfying (1.3.3). Then, under Assumption 1.3.1, ɛ ε p, χ ε p have the same regularity as V ε. Namely, the function (ε, t, x) ɛ ε p(t, x) belongs to C ([, 1] ε, C 1 W 1, ([, T ] R 2 )) and (ε, t, x, z) χ ε p(t, x, z) C ([, 1] ε, C 1 W 1, ([, T ] R 2 x, H 2 (, 1))) for any T >. Moreover, let Π ε p be the orthogonal projector on the p-th eigenspace and let t Π ε p and x Π ε p be their derivatives defined by the following commutators : t Π ε p = [ t, Π ε p], x Π ε p = [ x, Π ε p]. For any p 1 and T >, Π ε p is a self-adjoint operator on L(H) and the function (ε, t, x) Π ε p as well as their derivatives with respect to t and x are continuously bounded functions of (ε, t, x) [, 1] [, T ] R 2 valued in L(H). Proof. This is a standard result using the perturbation theory of linear operators (see for instance [14, 22]). We proceed now to the proof of the first point of Theorem Under Assumption and by (1.3.16), we have, up to extraction of subsequence, W ε (Ψ ε I, Ψ ε I) WI as ε

65 1.3. SEMICLASSICAL LIMIT AND PARTIALLY CONFINING POTENTIAL 65 in A H weak, where W I is a bounded measure with 2 2 hermitian, nonnegative matrix value. Moreover, in view of Lemma , Π ε p(, x)ϕ converges strongly to Π p (, x)ϕ as ε in A for every ϕ A. Therefore, Π ε p(, x)w ε (Ψ ε I, Ψ ε I)Π ε p(, x) Π p (, x)w I Π p (, x) in A weak and thanks to Lemma and the self adjointness of Π ε p, one obtains W ε (Π ε p(, x)ψ ε I, Π ε p(, x)ψ ε I) Π p (, x)w I Π p (, x) in A weak. Moreover, we have WI M b (R 4, H H) with WI (x, ξ x, z, z ) H 2 + (C) for all (x, ξ x, z, z ) R 4 [, 1] 2. Let F p,i = 1 1 W I (x, ξ x, z, z )χ p (, x, z)χ p (, x, z )dzdz, then F p,i (x, ξ x ) H 2 + (C) and Π p (, x)w I Π p (, x) = F p,i (x, ξ x )χ p (, x, z)χ p (, x, z ). For the second point of the theorem, let W ε p,q = W ε (Π ε pψ ε, Π ε qψ ε ). An analogous analysis using (1.3.6) and (1.3.16) shows that W ε p,q W p,q := Π p W Π q, as ε in L ((, T ), A ) weak for every T >, where W denotes the weak limit of W ε (Ψ ε, Ψ ε ) in L ((, T ), A ). It is a bounded measure belonging to L ((, T ), M b (R 4, H H)) with H + 2 (C) matrix value. It remains now to compute W p,q for every p, q N. Lemma let Ψ ε p = Π ε pψ ε. Then, Ψ ε p C (R +, H 1 (R 2, H)) C (R +, L 2 (R 2, H 2 ((, 1), C 2 )) C 1 (R +, L 2 (R 2, H)) and satisfies with iε t Ψ ε p = ε2 2 xψ ε p + ɛ ε pψ ε p + R ε p + επ ε pω W ε (t, x, z, i ε ). σ(ψ ε ) (1.3.24) Rp ε = iε( t Π ε p)ψ ε + ε2 2 xπ ε p x Ψ ε + ε2 2 div x(( x Π ε p)ψ ε ). In addition, for any T > lim sup sup Ψ ε N + p(t,.) =. (1.3.25) ε (,1] t [,T ] p N L 2 (R 2,H)

66 66 CHAPTER 1. SEMICLASSICAL ANALYSIS We refer also to [1] for the proof of this lemma. We deduce that, for every p, q N, W ε p,q satisfies t Wp,q ε = W ε ( t Ψ ε p, Ψ ε q) + W ε (Ψ ε p, t Ψ ε q) = iε ( W ε ( x Ψ ε 2 p, Ψ ε q) W ε (Ψ ε p, x Ψ ε q) ) i ( W ε (ɛ ε ε pψ ε p, Ψ ε q) W ε (Ψ ε p,ɛ ε qψ ε q) ) + W ε (( t Π ε p)ψ ε, Ψ ε q) + W ε (Ψ ε p, ( t Π ε q)ψ ε ) iε ( W ε ([ x, Π ε 2 p]ψ ε, Ψ ε q) W ε (Ψ ε q, [ x, Π ε q]ψ ε ) ) iw ε (Π ε pω W ε (t, x, z, i ε ). σ(ψ ε ), Ψ ε q) + iw ε (Ψ ε p, Π ε qω W ε (t, x, z, i ε ). σ(ψ ε )). (1.3.26) Multiplying this equation by ε and getting formally ε goes to zero, one gets (ɛ p ɛ q )W p,q =. Since ɛ p ɛ q if p q, one deduces that W p,q =, or Π p W Π q = for p q. This result can be checked rigorously in the distribution sense. That is to wit, W ε p,q if p q in D (R + t R 4 x,ξ x, H H). See [1] for details. Remark that we have, by construction of H H, W = p,q Π p W Π q and thus W = p N Π p W Π p := p N Π p W Π p. Let us now calculate the equation verified by Wp,p. For abuse of notations, we shall use Wp instead of Wp,p. For this we will perform the limit of (1.3.26) for p = q and for a given T >. Applying Lemma and with the boundedness of ɛ ε p, Π ε p and all its derivatives with respect to t and x, one can pass to the limit on all the terms of the right hand side of (1.3.26) in L ((, T ), A ) weak, to find ( ) Π p t Wp + ξ x x Wp x ɛ p ξx Wp Πp = iπ p [Wp, Ω(t, x, z, ξ x )]Π p. (1.3.27) See [1] for more details on the obtention of the left hand side of this equation. Let us just explain the ideas for the obtention of the right hand side. View Assumption 1.3.3, the properties of Π ε p (Lemma ) and Lemma 1.3.1, one can write W ε (Π ε pω W ε (t, x, z, i ε ) σ(ψ ε ), Ψ ε p) = Π ε pω W ε (t, x, z, ξ x ) σw ε (Ψ ε, Ψ ε )Π ε p + r ε where the both sides are bounded in L ((, T ), A ) and r ε tends to uniformly on [, T ] in A weak. Passing to the limit, we have W ε (Π ε pω W ε (t, x, z, i ε ) σ(ψ ε ), Ψ ε p) Π p Ω(t, x, z, ξx ) σw Π p = Π p Ω(t, x, z, ξx ) σw p Π p

67 REFERENCES 67 since W = q N W q = q N Π q W Π q. Using the same arguments, we have also W ε (Ψ ε p, Π ε pω W ε (t, x, z, i ε ) σ(ψ ε )) Π p Wp Ω(t, x, z, ξ x ) σπ p. We deduce then, by passing to the limit in (1.3.26) for p = q, that Wp satisfies (1.3.27) in the sense of distributions S ((, T ) R 4, H H) and that t Wp ε is bounded in L ((, T ), S (R 4, H H)). Thus, Wp ε is equicontinuous in t with values in S and converges locally uniformly with respect to t. One deduces that Wp C (R +, M b (R 4, H H)) and for the continuity of W stated in (1.3.9), it suffices to remark that (1.3.25) and (1.3.16) imply the uniform convergence of the series Wp ε C ([, T ], A ). p N Finally, since Wp = Π p Wp Π p then, Wp (t, x, ξ x, z, z ) = F p (t, x, ξ x )χ p (t, x, z)χ p (t, x, z ) with F p is an H 2 + (C) matrix valued function and (1.3.27) yields the first equation of (1.3.1). The third item of Theorem is a direct consequence of Proposition 1.3.9, of (1.3.9) and the uniform convergence of the series. The ε-oscillatory property is assured by the energy estimate (1.3.6) and the compactness property (1.3.18) holds true by setting P ε n = Π ε p p n and thanks to (1.3.25) and Lemma References [1] N. Ben Abdallah and F. Méhats, Semiclassical analysis of the Schrödinger equation with a partially confining potential, J. Math. Pures Appl. (9), 84 (25), pp [2] J. A. Bittencourt, Fundamentals of plasma physics, Pergamon Press, Oxford, [3] Y. A. Bychkov and E. I. Rashba, Oscillatory effects and the magnetic susceptibility of carriers in inversion layers, Journal of Physics C : Solid State Physics, 17 (1984), pp

68 68 REFERENCES [4] S. Datta and B. Das, Electronic analog of the electro-optic modulator, Applied Physics Letters, 56 (199), pp [5] G. Dresselhaus, Spin-orbit coupling effects in zinc blende structures, Phys. Rev., 1 (1955), pp [6] C. Fermanian-Kammerer and P. Gérard, Mesures semi-classiques et croisement de modes, Séminaire Équations aux dérivées partielles (Polytechnique) (1999-2), Art. No. 17, 13 p. [7], Une formule de Landau-Zener pour un croisement non dégénéré et involutif de codimension 3, C. R. Math. Acad. Sci. Paris, 335 (22), pp [8], A Landau-Zener formula for non-degenerated involutive codimension 3 crossings, Ann. Henri Poincaré, 4 (23), pp [9] C. Fermanian-Kammerer and P. Gerard, A Landau-Zener formula for two-scaled Wigner measures, in Dispersive transport equations and multiscale models (Minneapolis, MN, 2), vol. 136 of IMA Vol. Math. Appl., Springer, New York, 24, pp [1] C. Fermanian-Kammerer and P. Gérard, Two-scale Wigner measures and the Landau-Zener formulas, in Multiscale methods in quantum mechanics, Trends Math., Birkhäuser Boston, Boston, MA, 24, pp [11] A. Fert, A. Friederich, and al, Giant magnetoresistance of (1)fe/(1)cr magnetic superlattices, Phys. Rev. Lett., 61 (1988), pp [12] P. Gérard, Mesures semi-classiques et ondes de Bloch, (1991), pp. Exp. No. XVI, 19. [13] P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 5 (1997), pp [14] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, Heidelberg, [15] P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), pp [16] P. A. Markowich and N. J. Mauser, The classical limit of a self-consistent quantum-vlasov equation in 3D, Math. Models Methods Appl. Sci., 3 (1993), pp [17] P. A. Markowich, N. J. Mauser, and F. Poupaud, A Wigner-function approach to (semi)classical limits : electrons in a periodic potential, J. Math. Phys., 35 (1994), pp

69 REFERENCES 69 [18] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 199. [19] M. S. Mock, Analysis of mathematical models of semiconductor devices, vol. 3 of Advances in Numerical Computation Series, Boole Press, Dún Laoghaire, [2] A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, [21] Y. P. Raizer, Gas Discharge Physics, Springer, Berlin, [22] M. Reed and B. Simon, Methods of modern mathematical physics., Academic Press, New York, second ed., Functional analysis. [23] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 4 (1932), pp [24] I. Zutic, J. Fabian, and S. D. Sarma, Spintronics : Fundamentals and applications, Reviews of Modern Physics, 76 (24), p. 323.

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71 Chapter 2 On an hierarchy of kinetic and macroscopic models for spintronics Abstract In this chapter an hierarchy of kinetic and macroscopic models for semiconductor spintronics is presented and analyzed. We begin by presenting and studying the so called spinor Boltzmann equation. Starting with a rescaled version of the Boltzmann equation with different spin-flip and non spin-flip collision operators, different continuum models are derived. By comparing the strength of the spin-orbit scattering with the scaled mean free paths, we explain how some models existing in the literature (like the two-component models) can be obtained from the spinor Boltzmann equation. A new spin-vector drift diffusion model keeping spin relaxation and spin precession effects due to the spin-orbit coupling in semiconductor structures is derived and some of its mathematical properties are checked. Other spin-vector diffusif models like Spherical Harmonic Expansion (SHE), Energy-Transport and Drift-Diffusion with Fermi-Dirac statistics models are derived by means of the moment method and entropy minimization principle. 71

72 72 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS 2.1 Introduction The electrons are not only characterized by their electric charge but also by their intrinsic kinetic moment or the so called spin. The spintronics is a new booming domain of research which tries to control the spin and to use it as an additional degree of freedom or a new vector of information. Although the first researches in this domain were led essentially for structures based on magnetic multilayers [22], the spin dependent properties of the electron transport in semiconductors have recently attracted several attentions. There are typically two class of mechanisms [23] which make relax the spin vector in semiconductors. In one side, we have according to the Elliot-Yafet mechanism [49, 23] the instantaneous interactions of the particles with the crystal accompanied with reversal of the spin direction. They will be called the spin-flip interactions. These events are rare in semiconductors : typically less than one interaction up one thousand return the spin orientation [7]. The second category of mechanisms are relative to the effect on spin-orbit coupling of the asymmetry inversion that can exist in the system. They can be characterized by an effective magnetic field which makes precess the spin vector during the free path of the particles. The ability to control this effective magnetic field could allow spin control and manipulation in semiconductor heterostructures. Many theoretical models are used by the physical community for spin-polarized transport [35, 36, 38, 41, 42, 44, 46, 47, 49]. In microelectronics the drift-diffusion system is one of the most used model for modelling the transport of charged particles in semiconductors [32, 33], Plasma [5], Gas Discharges [39], etc. The drift-diffusion model, which describes the macroscopic behavior of the particles, is a very well suited model for numerical simulations. Two types of drift-diffusion approximations are essentially used in spintronics : the so called two-component drift-diffusion model and the spin polarization vector or density matrix based approximation. In the twocomponent description, the electrons are considered to be of two types, namely, having spin up or down. Each type of electrons is described by the usual driftdiffusion equation with additional terms related to sources and relaxation of the electron spin polarization, see [46, 47, 36]. The two-component model has been used initially for spin transport in ferromagnetic metals, it was used also in studies of propagation of spin-polarized electrons through a semiconductor region with variable level of doping [36]. In this kind of model, the mechanism of spin relaxation (such the spin-orbit interaction for instance) is not specified. The spin-vector (or density matrix) approach is a more general description in which the spin variable (the density or the distribution function for example) is a vector quantity and the mechanisms acting on the spin dynamics can be taken into account. The aim of this chapter is to derive and analyse macroscopic models for semi-

73 2.1. INTRODUCTION 73 conductor spintronics from the general spinor Boltzmann equation. The starting equation is the following scaled Boltzmann equation F ε t +1 ε (v xf ε x V v F ε ) = 1 ε Q(F ε )+ α [ ] i Ω(x, 2 ε 2 v) σ, F ε +Q sf (F ε ), (2.1.1) under the initial condition F ε (, x, v) = F in (x, v), (2.1.2) where ε > is a small positive parameter. It represents the scaled mean free paths. The parameter α is the scaled strength of the spin-orbit scattering. The operator Q is the collision operator and Q sf represents the spin-flip interactions (or interactions accompanied with reversal of spin s direction). We use the following relaxation time approximation of Q sf Qsf(F ) = tr(f )I 2 2F τ sf, (2.1.3) with τ sf > is the spin relaxation time. This operator makes relax, when τ sf goes to zero, the matrix distribution function to a scalar one. Since the spin-flip interactions are not frequent in semiconductor structures as we mentioned above, τ sf is not small and we assume that Q sf is a perturbation part of the collision operator. This is natural then to consider Q sf of order one in the diffusion scaling (2.1.1). The diffusion limit ε leads to macroscopic diffusion models (Drift-Diffusion, SHE, etc... ) according to the dominant scattering mechanisms. We refer to [1, 2, 12, 14, 19, 27, 37, 25, 43] for the rigorous derivation of macroscopic models from kinetic equations. Different kind of scattering operators are considered in this work. We consider first the collision operator for a Boltzmann statistics in the linear BGK approximation given by : Q(F ) = α(v, v )[M(v)F (v ) M(v )F (v)]dv. R 3 (2.1.4) The function M is the normalized Maxwellian M(v) = 1 e 1 (2π) 3 2 v 2, v R 3. (2.1.5) 2 The chapter is organized a follows. In the next section, we fix some notations used along this chapter. The study of the Boltzmann equation is carried out in Section 2.3. Existence and uniqueness of weak solutions of (2.1.1) is presented. It is a standard result of Boltzmann type equations. In the spinor Boltzmann description, the distribution function shall be a matrix valued function from R + R 6 into the space of 2 2 hermitian and positive matrices (H + 2 (C)). We prove that equation

74 74 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS (2.1.1) preserves the positivity and the self adjointness of the distribution function during the time. In other terms, the following maximum principle holds : if F in (x, v) H + 2 (C), (x, v) R 6 then, F ε (t, x, v) H + 2 (C), t > and (x, v) R 6. This means that if F ε satisfies (2.1.1), then (F ε ) is also a solution of (2.1.1). Moreover, if F in H + 2 (C) and if we decompose F ε into spin-dependent and spinindependent parts as F ε (t, x, v) = 1 2 f ε c (t, x, v)i 2 + f ε s (t, x, v) σ where f ε c is the charge distribution and f ε s is the spin distribution then, we have 1 f ε 2 c (t, x, v) f s (t, x, v) for every (t, x, v) R R 6. In the following sections we study the diffusion limit ε for different order of α with respect to ε. Section 2.4 is dedicated to the derivation of two-component models from the spinor Boltzmann equation. We begin by discussing what we call the decoherence limit. This limit corresponds to keeping ε constant and to taking α goes to +. It corresponds to taking a large spin-orbit coupling so that the ratio between the period of rotations (T ) induced by the spin-orbit effect and the used time scale (t) is small and goes to zero. This limit makes relax the spin part of the distribution function to the direction parallel to Ω. If the direction of Ω does not depend on v, a two-component kinetic model is obtained which yields two-component macroscopic model at the diffusion limit. In the next subsection, we study the diffusion limit of (2.1.1) with α = O( 1 ). This situation occurs in structures where the spin-orbit ε coupling is high such that the rotational period T is of the same order of the mean free path time τ and where T t = ε. Similarly, if the direction of Ω does not depend on v, the spin vector direction tends towards Ω and one gets at the limit a two component drift-diffusion model. However, if the direction of Ω depends on v, the spin information is lost at the limit. In other words, the spin vector relaxes towards zero and we obtain the standard scalar drift-diffusion model for the charge density (or the total density) used in microelectronics. This spin relaxation corresponds to the D yakonov-perel mechanism. It happens since the effective field changes frequently direction due to the numerous interactions that a particle undergo on its trajectory in the diffusion regime under investigation. In Section 2.5, we are interested by the derivation of general spin-vector driftdiffusion model with spin rotation and relaxation effects. Suppose first that α is of the same order as ε (α = O(ε)) and take α = ε for simplicity. This means that the order of the spin-orbit coupling is small in such a way that the rotation angle of the spin vector around the effective field Ω is small during the free paths of the

75 2.2. ASSUMPTIONS AND NOTATIONS 75 particles. In this case, F ε converges to N(t, x)m(v) (in the weak sense see Section 2.5) such that N is a positive hermitian matrix satisfying the following equation t N + div x (D( x N + x V N)) = i 2 [ H e σ, N] + tr(n)i 2 N τ sf, where D is a positive definite matrix and the effective field is an M-weighted averaging of Ω with respect to v : H e (x) = Ω(x, v)m(v)dv. R 3 Remark that if Ω is an odd vector with respect to v then H e = and no rotation effect appears at the limit. This is generally the case of the spin-orbit effective fields in semiconductor heterostructures (Rashba or Dresselhauss vectors). To keep trace of the spin-orbit interactions at the diffusion limit when Ω is an odd vector, one has to take a time scale such that α = O(1) with respect to ε. Applying this idea, a general spin-vector drift-diffusion model will be rigourously derived (Theorem ) and one of its main property to wit the conservation of the positivity and the self-adjointness of the density matrix during the time (maximum principle) will be checked (see Theorem 2.5.3). Following the same strategy, other two-component or spin-vector fluid models can be derived. Section 2.6 is dedicated to the derivation of spin-vector SHE (Spherical Harmonic Expansion) model when taking dominant elastic collisions. Finally, in Section 2.7 we discuss the derivation of Energy-Transport and Drift-Diffusion with Fermi-Dirac statistics models using the moment method and entropy minimization principle. We note that the method used is inspired from the works of P. Degond and C. Ringhofer [17, 18] for derivation of quantum hydrodynamic models. Other works on derivation and numerical study of quantum fluid models exist [15, 16, 13, 24]. 2.2 Assumptions and notations Let us begin by introducing some assumptions and notations. Assumption The cross-section, α(.,.), of the collision operator (2.1.4) belongs to W 1, (R 6 ) and is assumed to be symmetric and bounded from above and below : α 1, α 2 >, < α 1 α(v, v ) α 2, v, v R 3. Assumption For any fixed T >, the potential (t, x) V (t, x) is a non negative real function belonging to C 1 ([, T ], W 1, (R 3 )).

76 76 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS We will use M 2 (C) to denote the space of 2 2 complex matrices ; H 2 (C) denotes the subspace of hermitian matrices and H + 2 (C) the subspace of hermitian positive matrices. We will denote by. 2 and.,. 2 the Frobenuis norm and the associated Frobenuis inner product A, B 2 = R(A : B) = R( 2 A ij B ij ), A 2 2 = A, A 2 = i,j=1 2 A ij 2 where for z C, R(z) is the real part of z and for any two complex matrices A, B M 2 (C) A : B = i,j Definition We define the space L 2 M by L 2 M = {F = F (x, v) H 2 (C) such that A ij B ij. F (x, v) 2 2 R M 6 This is an Hilbert space equipped with the following scalar product i,j=1 dxdv < + }. (2.2.1) F, G 2 F, G M = R M dxdv, 6 and. M will denote the norm associated to.,. M. The same space with scalar valued functions will be denoted by L 2 M instead of L2 M. 2.3 Study of spinor Boltzmann type models The aim of this section is to study the properties of the spinor Boltzmann equation with the spin-orbit term. Existence and uniqueness of weak solution of (2.1.1) will be shown and some a priori estimates on the solution independent of the parameters α and ε will be given (Theorem 2.3.2). The contents of this first part review some well known results on linear Boltzmann type equations. We begin by defining the notion of weak solution of (2.1.1). Definition (weak solution). For a fixed time T >, a function F ε L 2 ([, T ]; L 2 M ) is called weak solution of (2.1.1) if it satisfies : T T F ε, t ψ 2 dtdxdv 1 F ε, v x ψ x V v ψ 2 dtdxdv = R ε 6 R 6 1 T Q(F ε ), ψ ε 2 2 dtdxdv + α T i R ε 6 R 2 [ Ω(x, v) σ, F ε ], ψ 2 dtdxdv 6 T + Q sf (F ε ), ψ 2 dtdxdv + F in, ψ() 2 dxdv, (2.3.1) R 6 R 6

77 2.3. STUDY OF SPINOR BOLTZMANN TYPE MODELS 77 for all ψ Cc 1 ([, T ) R 6 ; H 2 (C)). Theorem For all fixed ε >, α >, T, F in L 2 M and under Assumptions 2.2.1, 2.2.2, the model (2.1.1)-(2.1.2) admits a unique weak solution F ε C ([, T ]; LM 2 ) satisfying F ε (t) L 2 M C, N ε L 2 t,x ([,T ] R 3 ) C t >, (2.3.2) F ε P(F ε ) 2 L 2 t (L2 M ) Cε2, (2.3.3) where C > is a general constant independent of α and ε. Here, P is the orthogonal projection on Ker(Q) which satisfies : P(F ε ) = N ε M with N ε := R 3 F ε dv. In addition the following maximum principle holds : if F in (x, v) H + 2 (C), (x, v) R 6 then F (t, x, v) H + 2 (C) t [, T ], (x, v) R 6. The next proposition summarizes some fundamental properties of the collision operator (2.1.4). Since it acts only on the speed variable v, t and x are considered as a parameters and are omitted in the next proposition for the sake of simplicity. Proposition (Properties of the collision operator (2.1.4)). Under Assumption 2.2.1, the collision operator given by (2.1.4) satisfies the following properties. (i) For all F L 2 M, we have the mass conservation : Q(F )(v)dv =. R 3 (ii) The mapping Q : L 2 M L2 M is a linear, continuous, selfadjoint and nonpositive operator. (iii) The kernel of Q is Ker(Q) = {F L 2 M, such that N H 2 (C), F (v) = NM(v)}. (iv) Let P be the orthogonal projection on KerQ, then we have the following coercivity inequality Q(F ), F M α 1 F P(F ) 2 M. (2.3.4) (v) The range of Q, R(Q), is a closed subset of L 2 M such that { ( R(Q) = Ker(Q) = F L 2 M, such that F (v)dv = R 3 )}.

78 78 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS Proof. The first point (i) is a direct consequence of Assumption The linearity of Q is obvious. Let F, G L 2 M, we have ( [ F (v Q(F ), G M = R α(v, v )M(v ) )M(v) R M(v 3 ) F (v) ] ) G(v) : M(v) M(v) dvdv. (2.3.5) Using the symmetry of the cross-section α, one can write ( Q(F ), G M = 1 [ F (v 2 R α(v, v )M(v ) )M(v) R 3 M(v ) F (v) ] [ ) G(v : ) M(v) M(v ) G(v) ]dvdv. M(v) (2.3.6) This implies that Q is selfadjoint and negative operator on L 2 M. By Assumption and the fact that F (v) R 3 2 dv F M, (2.3.5) gives Q(F ), G M C F M G M, which yields the continuity of Q. The third point is obvious using the equality (2.3.6). Namely, the following equivalences hold : (F KerQ) ( Q(F ), F M = ) ( N H 2 (C), F (v) = NM(v)). The orthogonal of KerQ is then given by the set : (KerQ) = { F L 2 M, such that R 3 F (v)dv = ( )}. (2.3.7) Let now F L 2 M and let G = F PF where PF is the orthogonal projection of F on KerQ. Then, using the self-adjointness of Q, we have Q(F ), F M = Q(G), G M. Since G (KerQ) and with Assumption we get Q(F ), F M = 1 α(v, v )M M G 2 M G M 2 2dvdv G 2 2 α 1 R M dv = α 1 F PF 2 M. 3 For the last point, since Q is selfadjoint we have R(Q) = (KerQ). It remains to show that R(Q) is closed. Indeed, let (H p ) p 1 R(Q) be a sequence converging to H L 2 M with respect to. M. Writing H p = Q(F p ) = Q(G p ), where G p = F p PF p, we have α 1 G p G q 2 M Q(G p ) Q(G q ), G p G q M H p H q M G p G q M.

79 2.3. STUDY OF SPINOR BOLTZMANN TYPE MODELS 79 Therefore, (G p ) p is a cauchy sequence in L 2 M. There exists G L2 M such that (G p) p converges to G. By the continuity of Q and the uniqueness of the limit, we deduce that H = Q(G). Proof of Theorem This is a standard result of the Boltzmann type equations using a fixed point theorem and the characteristics method. Let us recall the main ideas of the proof. For notational simplicity, we omit the presence of α and ε (supposing α = ( ε = 1). We decompose ) the collision operator as, Q = Q + + Q when Q + (F ) = α(v, v )F (v )dv M(v) and Q (F ) = ν(v)f with ν(v) = R 3 α(v, v )M(v )dv. R 3 Lemma Let L = t + v x x V v, then for any F in L 2 M and S C ([, T ]; L 2 M ), the problem L(F ) Q (F ) i 2 [ Ω σ, F ] Q sf (F ) = S F (t = ) = F in (2.3.8) admits a unique weak solution satisfying the following estimate F (t) L 2 MV F in L 2 MV + t S(s) L 2 MV ds (2.3.9) with M V = 1 e 1 (2π) 3 2 v 2 +V. In addition, the maximum principle holds which means 2 that if F in, S are two hermitian positive matrices functions then the solution of (2.3.8), F, is also in H + 2 (C), (t, x, v) [, T ] R 6. Proof. Using the decomposition of any matrix A H 2 (C) into spin-dependent and spin-independent parts as A = A c I 2 + A s σ, H 2 (C) is equivalent to R 4 (we use the identification A (A c, A s ) for any A H 2 (C)). Any function F L 2 M is seen as a function in (L 2 M )4. Let F in (Fc in, F s in ) and S (S c, S s ), then problem (2.3.8) is equivalent to finding F (F c, f s ) in C ([, T ], (L 2 M )4 ) such that { L(F ) + C(v)(F ) + A(x, v)(f ) = S (2.3.1) F (t = ) = F in. The function C(v) is the matrix associated to the (Q + Q sf ) operator given by ( ) ν(v) C(v) = (ν(v) + 1, τ sf )I 3 I 3 denotes the identity matrix of order three. For all (x, v) R 6, A(x, v) is an antisymmetric 4 4 matrix associated to the spin-orbit anti-joint operator ( i 2 [ Ω σ,.]).

80 8 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS ( ) It has the following form : A(x, v) = with 3 3 antisymmetric I(x, v) matrix function I(x, v). Assuming first that F in and S are regulars, then, thanks to the characteristics method, the solution of (2.3.1) is given by the following integral formula ( F (t, z) = exp + t t ) [C(Z(s; z, t)) + A(Z(s; z, t))]ds F in ( t ) exp [C(Z(τ; z, t)) + A(Z(τ; z, t))]dτ S(Z(s; z, t))ds, s (2.3.11) where z = (x, v) is the variable in the phase space. For any z = (x, v) R 6, t [, T ], the characteristics Z(s; z, t) := (X (s; z, t), V(s; z, t)) satisfy the following differential system dx ds = V(s), dv = E(X (s)), ds E = xv, (X (t; z, t), V(t; z, t)) = (x, v). In this case, F given by (2.3.11) is also the weak solution of (2.3.1). By scalar multiplying (2.3.8) by F M V and integrating with respect to t, x, and v (i.e. rigourously choosing it as a test function in the weak formulation), one can find estimate (2.3.9) since Q and Q sf are two negative operators on H 2 (C) and i 2 [ Ω σ, F ], F 2 =. Furthermore, using the integral formula (2.3.11), one can verify straightforwardly the maximum principle. If F in and S belong to L 2 M and C ([, T ], L 2 M ) respectively, these results remain correct and can be obtained by regularization technics. The proof of Lemma is achieved. To continue the proof of Theorem 2.3.2, we define χ : L 2 M C ([, T ], L 2 M) C ([, T ], L 2 M) (F in, S) χ(f in, S) = F, where χ(f in, S) := F is the unique weak solution of (2.3.8) with initial condition F in and source term S. The mapping χ is linear, continuous (in view of estimate (2.3.9)) and satisfies the maximum principle. Moreover, a function F satisfies (2.1.1)-(2.1.2) if and only if F = χ(f in, Q + (F )) which is equivalent to say that F is a fixed point of Θ : F C ([, T ], L 2 M) χ(f in, Q + (F )) C ([, T ], L 2 M). To show that Θ admits a unique fixed point, let us define, for every δ, the norm. δ on C ([, T ], L 2 M ) by

81 2.4. TWO-COMPONENT MODELS 81 ) F δ = sup (e δt F L. All these norms are equivalent to the usual one. 2M (for t [,T ] δ = ), and one can choose δ small enough so that Θ forms a contracted mapping on (C ([, T ], L 2 M ),. δ). As a conclusion, problem (2.1.1)-(2.1.2) admits a unique weak solution satisfying the maximum principle (since Q + (F ) conserves the positivity of F and by Lemma 2.3.4) and we have T F F (t) L 2 MV F in L 2 MV + Q(F ), 2 dsdxdv t [, T ]. R M 6 V Furthermore, taking into account the parameters α and ε, the weak solution, F ε, of (2.1.1)-(2.1.2) satisfies F (t) L 2 MV F in L 2 MV + 1 T F Q(F ), ε 2 2 dsdxdv t [, T ]. R M 6 V F Finally, since Q(F ), 2 dv, one deduces estimates (2.3.2) and the balance R M 3 V deviation inequality (2.3.3) follows from the coercivity inequality (2.3.4). 2.4 Two-component models This section is concerned with the derivation of two-component kinetic and macroscopic models from the general spinor kinetic equation Decoherence limit We explain in this subsection how the spin-orbit interactions act on the distribution function when the order of this coupling becomes large. We assume that the period of rotation T of the spin vector distribution part around the effective field Ω is small in front of the time scale t of the problem. The decoherence limit is the limit η = T t. This makes relax the spin part of the distribution function F η of (2.4.1) towards the effective field line. This is the subject of the next proposition. Proposition Assume that Ω satisfies Assumption 2.4.3, F in L 2 M and that Assumptions 2.2.1, hold. Let T > and F η L 2 ([, T ], L 2 M ) be the weak solution of t F η + v x F η x V v F η = Q(F η ) + i 2η [ Ω σ, F η ] + Q sf (F η ) (2.4.1) with F η (, x, v) = F in (x, v). Then, when η goes to, F η tends to F such that F (t, x, v) = f c(t, x, v) I 2 +f s (t, x, v) ω(x, v) σ with f c and f s belong to L 2 ([, T ], L 2 M 2 ). In addition, the charge and spin distribution functions, f c and f s, satisfy weakly

82 82 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS t f c + v x f c x V v f c = Q(f c ) (2.4.2) t f s + v x f s x V v f s = Q (f s ω) ω 2 f s τ sf (2.4.3) and F (, x, v) = F in (x, v) where for any (x, v) R 6, ω(x, v) is the unit vector of the effective field line. Proof. Equation admits a unique weak solution F η L 2 ([, T ]; L 2 M ) such that (F η ) η is bounded with respect to η (see Section 2.3 for details). There exists F L 2 ([, T ]; L 2 M ) such that F η F weakly in L 2 ([, T ]; L 2 M ). This implies that i[ Ω σ, F η ] is also bounded in L 2 ([, T ]; L 2 M ) with respect to η and i[ Ω σ, F η ] i[ Ω σ, F ]. Multiplying the weak formulation of (2.4.1) by η and taking η tends to zero, we get i[ Ω σ, F ] =. This implies that the spin part of F is parallel to Ω i.e. there exist f c and f s in L 2 ([, T ]; L 2 M ) such that F = f c 2 I 2 + f s ω σ. Decomposing (2.4.1) into charge and spin parts by setting F η = f η c 2 + f η s σ, one has t f η c + v x f η c x V v f η c = Q(f η c ) t f η s + v x f η s x V v f η s = Q(f η s ) 1 η Ω f η s 2 f η s τ sf. (2.4.4) The weak limit of the first equation is (2.4.2). Taking the scalar multiplication of (2.4.4) with ω and passing to the limit weakly in L 2 M ([, T ] R6 ) one finds (2.4.3). Remark If we suppose that the direction of Ω, ω, does not depend on v then, we obtain at the decoherence limit a two-component kinetic model describing the evolution of spin-up and spin-down distribution functions f and f. These functions are nothing but the eigenvalues of F choosing such that : f = f c + f s and f = f c f s. If f c, f s satisfy (2.4.2)-(2.4.3), then f and f satisfy the following twocomponent kinetic model t f + v x f x V v f = Q(f ) + f f, τ sf t f + v x f x V v f = Q(f ) + f f τ sf, (2.4.5) subject to the initial conditions : f () = f in c 2 + f in s ω and f () = f in c 2 f in s ω, ( where fin c and f in s are the charge and spin parts of F in F in = f ) in c 2 I 2 + f in. σ s. The model (2.4.5), leads then to a two-component macroscopic model in this case (the case when the effective field direction is independent on v).

83 2.4. TWO-COMPONENT MODELS 83 In the next subsection, we will pass from a spinor Boltzmann equation to a 2- component macroscopic (drift-diffusion) model. We will see also that this asymptotic is possible if the effective field line does not depend on v and it corresponds to taking a diffusion limit of the spinor Boltzmann equation with high spin-orbit coupling (case α = O( 1 ε )) Diffusion limit with strong spin-orbit coupling : twocomponent Drift-Diffusion model This part is intended to the study of the diffusion limit (ε ) of (2.1.1) when α = O( 1 ) with respect to ε. As mentioned in the introduction, the macroscopic ε model, one obtains in this case, changes wether the direction of Ω depends on v or not. We will see that if the effective field Ω does not change direction with v this scaling gives at the limit a two-component Drift-Diffusion model. For simplicity, we suppose that α = 1 and the starting equation is then ε F ε t + 1 ε (v xf ε x V v F ε ) = 1 { Q(F ε ) + i } ε 2 2 [ Ω σ, F ε ] + Q sf (F ε ), (2.4.6) with initial condition (2.1.2). The operators Q and Q sf (2.1.3) respectively. We will use the following form of Ω. are given by (2.1.4) and Assumption We assume that Ω belongs to C 2 (R 6, R 3 ) and is given by Ω(x, v) = λ(x, v) ω(x, v), such that ω(x, v) = 1, (x, v) R 6 where λ and ω are two regular respectively scalar and vectorial functions. In addition, we suppose that the following polynomially controls at infinity with respect to v hold C 1 (1 + v ) m λ(x, v) C 2 (1 + v ) m (2.4.7) ηω + ηη 2 ω(x, v) C(1 + v )m (2.4.8) η {x i,v i } η,η {x i,v i } for C 1 >, C 2 >, C > and m N. Definition We denote by Q SO the following unbounded operator on L 2 M Q SO = Q + i 2 [ Ω σ,.] (2.4.9) defined on the following domain { D(Q SO ) = F L 2 M / i[ Ω } σ, F ] L 2 M { = F = tr(f ) I 2 + f 2 s σ L 2 M / Recalling that L 2 M denotes the space given by Definition } Ω fs L 2 M. (2.4.1)

84 84 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS In view of the properties of the collision operator listed in Proposition 2.3.3, we present in the following proposition some important properties of Q SO. Proposition Under Assumptions 2.2.1, 2.4.3, the unbounded operator (Q SO, D(Q SO )) given by (2.4.9)-(2.4.1) satisfies the following properties. 1. It is a maximal monotone operator on L 2 M. 2. Let Ker(Q SO ) be the nul space of Q SO, then we have the following characterization : { ker(q SO ) = F = N(x)M(v)/ N = N c 2 I 2 + N s σ L 2 (R 3, H 2 (C)) { and N if ω depends on v s = n s (x) ω if ω = ω(x) independent on v. } (2.4.11) 3. The range of Q SO is given by { Im(Q SO ) = G = g c 2 I 2 + g s σ L 2 M / g c dv = R ( ) 3 } and g s dv ω = if ω does not depend on v R 3 (2.4.12) Proof. 1. The adjoint of Q SO is given by defined on D(Q SO ) = D(Q SO). Indeed, by definition Q SO = Q i 2 [ Ω σ,.] (2.4.13) D(Q SO) = { F L 2 M/ G F, Q SO (G) M is a bounded operator on D(Q SO ) }. For every F D(Q SO ), G D(Q SO), we have by the self-adjointness of Q This implies F, Q SO (G) M = Q(F ) i 2 [ Ω σ, F ], G M. (2.4.14) i 2 [ Ω σ, F ], G M = Q(F ), G M F, Q SO (G) M for every G D(Q SO ). We deduce that for F D(Q SO ), i 2 [ Ω σ, F ] is a linear and continuous operator on D(Q SO ) which is dense in L 2 M. It can be then prolonged to a linear continuous operator on L 2 M which implies that (since L2 M is an Hilbert space) i 2 [ Ω σ, F ] L 2 M and thus F D(Q SO ) if F D(Q SO ). The reciprocal inclusion (D(Q SO ) D(Q SO )) is obvious and from (2.4.14), one deduces that Q SO is given

85 2.4. TWO-COMPONENT MODELS 85 by (2.4.13) on D(Q SO ). In other side, i[ Ω σ, F ], F M = for every F L 2 M. Then, since Q is a non positive operator, we have Q SO (F ), F M = Q SO(F ), F M = Q(F ), F M and the operators Q SO and Q SO are monotones. Moreover, D(Q SO) is dense in L 2 M and the graph of Q SO, G(Q SO ), is closed. Indeed, let (F n, Q SO (F n )) n N such that F n D(Q SO ) be a sequence in G(Q SO ) converging to (F, G) in (L 2 M )2. We have to prove that F D(Q SO ) and G = Q SO (F ). For every H D(Q SO ), one has F n, Q SO(H) M = Q SO (F n ), H M. By passing to the limit, n +, one gets F, Q SO(H) M = G, H M for every H D(Q SO ) and since D(Q SO ) = L 2 M, we deduce that F D(Q SO) and Q SO (F ) = G. As a consequence, Q SO is a densely defined closed operator such that Q SO and Q SO are monotones. It is then a maximal monotone operator on L2 M. 2. Let F Ker(Q SO ), we have Q(F ) + i 2 [ Ω σ, F ] =. (2.4.15) Taking the scalar product with F in L 2 M, one gets Q(F ), F M =. This implies that Q(F ) = and F = N(x)M(v) such that N L 2 (R 3, H 2 (C)) (see Proposition 2.3.3). Writing N = N c 2 I 2 + N s σ and inserting it in (2.4.15), we obtain Ω N s =. (2.4.16) One can deduce simply that N s = if Ω changes direction with v and if not, the vector N s is parallel to Ω. 3. Since Q SO is a closed and densely defined operator on L 2 M, we have Im(Q SO ) = (KerQ SO). Moreover, we have Ker(Q SO ) = Ker(Q SO) and it is simple to verify that the orthogonal of Ker(Q SO ) is nothing else but the set given (2.4.12). This implies that Im(Q SO ) Ker(Q SO ). In other side, let G = g c 2 I 2 + g s σ ker(q SO ) which ( ) means that g c dv = and g s dv ω = if ω = ω(x) does not depend on R 3 R 3 v. Viewing the properties of the collision operator Q (Proposition 2.3.3), there is a

86 86 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS unique function f c L 2 M (R6 ) such that f c (x, v)dv = and Q(f c ) = g c. It remains to verify the existence of a unique f s (L 2 M (R6 )) 3 such that ( fs dv) ω = R 3 R if ω does not depend on v and 3 Q( f s ) λ( ω f s ) = g s. Since Q SO is a maximal monotone operator, then δ >, δid Q SO is surjective, where Id denotes the identity operator on L 2 M. There exists a vector function f δ s such that f δ s σ D(Q SO ) for any δ > and (δid Q SO )( f δ s σ) = g s σ. Then, δ f δ s Q( f δ s ) + λ ω f δ s = g s, for all δ >. We have to prove now that the sequence ( f δ s ) δ is bounded in (L 2 M (R6 )) 3. We argue by contradiction and assume the existence of a subsequence denoted also by ( f δ s ) δ such that f δ s δ f δ s = f s δ, we have fs δ δ + with. is the norm in (L 2 M (R6 )) 3. Denoting by f δ s Q( f δ s) + λ ω f δ s = g s f δ s, (2.4.17) and f δ s = 1. Then, by passing to the limit weakly in (L 2 M )3, we have f δ s f s in (L 2 M )3 such that Q( f s ) + λ ω f s = which implies that f s = if ω depends on v and f s = n s (t, x) ω(t, x)m if not. Moreover, if ω is independent on v, g s satisfies ( g s dv) ω =. Then, integrating R 3 (2.4.17) with respect to v and multiplying by ω, the same condition is also satisfied by ( f δ s) : f δ sdv ω = for every δ >. Getting δ, one deduces that n s = R 3 ( fs dv) ω =. Hence, f δ s f s =. In other side, let us show that f δ s f s strongly R 3 in (L 2 M )3. This implies that f s = 1, since f δ s = 1 δ > which is in contradiction with f s =. Indeed, rewriting equation (2.4.17) as follows (δ + ν(v)) f δ s + λ ω f δ s = g δ s + Q + ( f δ s), (2.4.18) with ν(v) = α(v, v )M(v )dv, Q + ( f δ s) = α(v, v ) f δ s(v )dv M(v) and g δ s = R 3 R g 3 s f. The solution of (2.4.18) can be computed explicitly. Indeed, without loss of s δ generality, assume that ω = (w 1, w 2, w 3 ) is such that w 3, ω = 1, and complete it to an orthonormal basis of R 3 : (ε 1, ε 2, ω). The change-of-basis matrix, P, from

87 2.4. TWO-COMPONENT MODELS 87 the standard euclidian basis to the new one is an orthogonal matrix ( t P P = I 3 ) given by P = 1 (w w 2 3) 1 2 w 3 w 1 w 2 w 1 (w w 2 3) 1 2 w w 2 3 w 2 (w w 2 3) 1 2 w 1 w 2 w 3 w 3 (w w 2 3) 1 2. (2.4.19) Let F δ s = t P f δ s be the new coordinates of f δ s in the new basis (ε i ) i. Then, it satisfies the following equation N δ ( F δ s) = t P ( h δ s), (2.4.2) where h δ s = g δ s + Q + ( f δ s) is the second member of (2.4.18) and 1 λ λ(x, v) N δ = (δ + ν(v)) λ 1, λ(x, v) = δ + ν(v). (2.4.21) 1 It is simple to verify that h δ s converges strongly in (L 2 M )3 to Q + ( f s ). Moreover, N δ is invertible and N 1 δ = 1 δ + ν(v) 1 1+ λ 2 λ 1+ λ 2 λ 1 1+ λ 2 1+ λ 2 1 It is a bounded matrix with respect to δ uniformly with respect to (x, v) : N 1 δ 2 2 if the cross section α(v, v ) satisfies Assumption As a conclusion, we δ + α 1 have f δ s = P N 1 δ tp ( h δ s) with ( h δ s) δ is a strongly convergent sequence in (L 2 M )3 and N 1 δ is a uniformly bounded matrix with respect to δ. Then, ( f δ s) δ converges strongly in (L 2 M )3. The proof of the proposition is completed. The following lemma follows from the last proposition.. Lemma There exists a unique χ s (L 2 M) 3 satisfying Q( χ s ) + λ( ω χ s ) = v x ωm (2.4.22) under the following condition ( ) χ s (x, v)dv ω =, x R 3. (2.4.23) R 3

88 88 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS Theorem Let T >, F in L 2 M and assume that Assumptions 2.2.1, 2.2.2, hold and that the direction of the effective field ω is independent on v. Then, the sequence of weak solutions, (F ε ) ε>, of (2.4.6)-(2.1.2) converges weakly in L 2 ([, T ]; L 2 M ), when ε goes to zero, to N(t, x)m(v) with N L2 ([, T ] R 3, H + 2 (C)) and such that N(t, x) = n c(t, x) I 2 + n s (t, x) ω(x) σ (2.4.24) 2 (the spin part of N is parallel to ω). In addition, the spin-up and spin-down densities, n = n c + n s and n = n c n s satisfy the following two-component drift-diffusion model t n div x (D 1 ( x n + x V n )) = n n τ(x) (2.4.25) t n div x (D 1 ( x n + x V n )) = n n τ(x) where D 1 is a symmetric positive definite matrix given by (2.5.1). We obtain at the limit a modified spin relaxation time given by τ(x) = 2τ sf 2 + τ sf χ(x) where χ(x) is a positive function Q( χ s ) χ s χ(x) = dv (2.4.26) R M 3 with χ s satisfies (2.4.22). Proof. With estimate (2.3.2), there exist F L 2 ([, T ], L 2 M ) and N L2 ([, T ] R 3 ; H 2 + (C)) such that F ε F and N ε N in the corresponding spaces and N = F dv (since N ε = F ε dv, ε > ). Multiplying (2.4.6) by ε 2 and passing R 3 R to the weak 3 limit ε, one gets in the distribution sence Q(F ) + i [ ω σ, F ] =. 2 Since ω is independent on v and with (2.4.11), F = N(t, x)m(v) such that the density matrix N can be written as (2.4.24). Let N ε = nε c 2 I 2 + n ε s σ and F ε = fc ε 2 I 2 + f s ε σ with n ε c = fc ε dv and n ε s = f ε s dv. Then, n ε c n c in L 2 ([, T ] R 3 ) R 3 R 3 and n ε s n s ω in (L 2 ([, T ] R 3 )) 3 (or n ε s ω n s ) where n c and n s are the charge and spin parts of N (2.4.24). Integrating equation (2.4.6) with respect to v, one obtains the following continuity equations t n ε c + div x jc ε = t n ε s + x Js ε = 1 ( Ω ε f 2 s ε )dv 2 nε s (2.4.27) R τ 3 sf

89 2.4. TWO-COMPONENT MODELS 89 where the charge and spin currents, jc ε ans Js ε, are given by jc ε = 1 vfc ε dv Js ε = 1 (v f ε R ε s ε )dv. 3 R 3 These continuity equations can be obtained weakly by taking test functions constants with respect to v in the weak formulation (2.3.1) (this choice of test functions is possible, see the next section). Moreover, using estimate (2.3.3), there is R ε in L 2 ([, T ]; L 2 M ) bounded with respect to ε such that F ε = N ε M + εr ε. In terms of spin and charge parts, we have f s ε = n ε sm + ε r s, ε r ε s L 2 t ((L 2 M )3 ) C, f ε c = n ε cm + εr ε c r ε c L 2 t (L 2 M ) C (2.4.28) where C > is a general constant independent of ε. Thus, jc ε = vrcdv ε and (jc) ε ε R 3 is bounded with respect to ε in L 2 ([, T ] R 3 ). It converges weakly to a function j c in L 2 ([, T ] R 3 ) and by passing to the limit on the first equation of (2.4.27), we have t n c + div x j c =. (2.4.29) Moreover, multiplying the second equation of (2.4.27) by ω, we get t ( n ε s ω) + div x (Js ε ( ω)) = Js ε : ( x ω) 2 nε s (2.4.3) τ sf where Js ε ( ω) = 1 (v f ε s ε )( ω)dv = 1 v( f R ε s ε ω)dv = v( r s ε ω)dv bounded 3 R 3 R 3 with respect to ε. Let us denote by j s the weak limit of Js ε ( ω) in L 2 ([, T ] R 3 ). Besides, let S ε := Js ε : ( x ω). Then, S ε = 1 (v x ω) ε f s ε dv which is also R 3 bounded with respect to ε in L 2 ([, T ] R 3 ) and converges weakly to a certain function S L 2 ([, T ] R 3 ). By passing to the weak limit ε, (2.4.3) yields the following continuity equation t n s + div x (j s ) = S 2n s τ sf. (2.4.31) To close this equation, one has to express j s and S according to n s. For this, taking the Frobenius inner product of (2.4.6) with θ 1 ω σ M where θ 1 is given by (2.5.8) and integrating with respect to v yields Js ε (w) = ε t ( f s ε ω) θ 1 R M dv v ( x + x V )( n ε s ω)θ 1 dv n ε s (v x ω)θ 1 dv 3 R 3 R 3 ε (v x x V v )( r s ε ω) θ 1 R M dv ε ( f s ε ω)θ 1 τ 3 sf R M dv, 3

90 9 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS up to straightforward computations using the self-adjointness of the collision operator Q and the expansion of f ε s around the equilibrium (2.4.28). Taking ε goes, to zero one obtains Js ε ( ω) j s = D 1 ( x n s + x V n s ) (v x ω) ωn s θ 1 dv R 3 = D 1 ( x n s + x V n s ) (since ω = 1) (2.4.32) with D 1 = (θ 1 v)dv. To rigourously find the relation between j s and n s, one R 3 has to use the weak formulation of (2.4.6) with θ 1 ω σ M φ(t, x), φ C1 c ([, T ) R 3 ), as test function and passing then to the limit (see the next section for details). A similar computation gives also j c = D 1 ( x n c + x V n c ). (2.4.33) Finally, we shall express the limit of S ε := 1 (v x ω) ε f s ε, S, in terms of n s. R 3 Taking the inner product of (2.4.6) with χ s σ M, where χ s satisfies (2.4.22)-(2.4.23), and integrating with respect to v, one obtains S ε = ε tf ε s χ s R M dv + (v x n ε s + v x V n ε s) χ s dv 3 R 3 + ε (v x x V v ) r s ε χ s R M dv + ε f ε s χ s τ 3 sf R M dv. 3 By passing to the limit ε, S = v x (n s ω) χ s dv + v x V n s ( ω χ s )dv. (2.4.34) R 3 R 3 This limit can be rigourously verified by taking χ s σ M φ(t, x), with φ C1 c ([, T ) R 3 ), as test function in (2.3.1). This choice is valid since χ s is polynomially increasing at infinity with respect to v (see Lemma 2.4.8). Moreover, multiplying (2.4.22) M by ω, we have Q( χ s ω) = with χ s ωdv = which implies that χ s ω =. In R 3 addition, if we multiply (2.4.22) by χ s and integrate with respect to v, we get M Q( χ s ) χ s (v x ω) χ s dv = dv = χ(x). R 3 R M 3 Consequently, the charge and spin densities n c and n s satisfy t n c div x (D 1 ( x n c + n c x V )) = which yields (2.4.25). t n s div x (D 1 ( x n s + n s x V )) = 2n s τ sf χ(x)n s

91 2.5. A GENERAL SPIN-VECTOR DRIFT-DIFFUSION MODEL 91 Lemma Let χ s be the solution of (2.4.22)-(2.4.23). Then under Assumption and Assumption 2.4.3, one has χ s M C(1 + v )m+1, η {x i,v i } with C is a general positive constant and m N. Proof. Rewriting equation (2.4.22) as η χ s M C(1 + v )m, ν(v) χ s + λ( ω χ s ) = (v x ω Q + ( χ s ))M(v) (2.4.35) with ν(v) = α(v, v )M(v )dv, Q + ( χ s ) = R 3 α(v, v ) χ s (v )dv and applying the R 3 same computations we have made for resolving equation (2.4.18), one finds The matrix P is given by (2.4.19) and N 1 = 1 ν χ s M = P N 1 tp (v x ω Q + ( χ s )) λ 2 λ 1+ λ 2 λ 1 1+ λ 2 1+ λ 2 1 λ(x, v), λ(x, v) =. ν(v) The matrices P and N 1 are uniformly bounded with respect to (x, v), P 2 = 3 and N α 1 (with Assumption 2.2.1). Therefore, using Assumption 2.4.3, we deduce that χ s M C(1 + v )m+1. Similarly, by differentiating (2.4.35) with respect to x or v, one can obtain the second estimates on η χ s M. 2.5 A general spin-vector Drift-Diffusion model This section is concerned with the diffusion limit when the spin-orbit coupling is of order one with respect to ε (α = O(1)). This scaling is useful to get a spin vector continuum model with rotation effects when the effective field of the spinorbit coupling is odd with respect to v. Here we take a general effective field Ω ε as follows Ω ε (x, v) = 1 ε Ω o (x, v) + Ω e (x, v), (2.5.1) where Ω o is odd with respect to v and Ω e is even with respect to v. For instance, Ω o can be the effective magnetic field following from the spin-orbit interactions (Rashba [1], Dresselhauss [2]) or the odd part of an applied magnetic field and Ω e

92 92 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS can represent the even part of an applied field. The starting scaled spinor Boltzmann equation is F ε t + 1 ε (v xf ε x V v F ε ) = 1 ε 2 Q(F ε )+ i 2 [ Ω ε (x, v) σ, F ε ]+Q sf (F ε ), (2.5.2) with the initial condition (2.1.2) and the operators Q, Q sf are respectively given by (2.1.4) and (2.1.3). Let us rewrite the weak formulation of (2.5.2). A function F ε L 2 ([, T ]; L 2 M ) is called weak solution of (2.5.2) if it satisfies : T T F ε, t ψ 2 dtdxdv 1 F ε, v x ψ x V v ψ 2 dtdxdv = R ε 6 R 6 1 T T Q(F ε ), ψ ε 2 2 dtdxdv + i R 6 R 2 [ Ω ε (x, v) σ, F ε ], ψ 2 dtdxdv 6 T + Q sf (F ε ), ψ 2 dtdxdv + F in, ψ() 2 dxdv, (2.5.3) R 6 R 6 for all ψ C 1 c ([, T ) R 6 ; H 2 (C)). Assumption We assume that Ω o (x, v) and Ω e (x, v) are respectively two regular odd and even vectors with respect to v. In addition, we suppose that Ω o is compactly supported with respect to x and there exist a constant C > and m N such that Ω o (v) + η {x i,v i } η Ωo (v) C (1 + v ) m. (2.5.4) The main results of this section is stated in the following two theorems. Theorem Let T >, F in L 2 M and assume that Assumption 2.2.1, Assumption and Assumption hold. Let for all ε > F ε C ([, T ]; L 2 M ) be the weak solution of (2.5.2)-(2.1.2). Then, the matrix density N ε := F ε (t, x, v)dv R 3 converges weakly in L 2 ([, T ] R 3, H 2 (C)) to N which satisfies the following equation t N div x {D 1 ( x N + N x V ) id 2 [ σ, N]} = i 2 [ Ω σ, N] + (D 4 tr(d 4 ))( N s ) σ + Q sf (N) (2.5.5) with initial condition N(, x) = R 3 F in (x, v)dv, and where N s is the spin density part of N. In addition, if we decompose N as : N = N c 2 I 2 + N s σ, then the charge and spin densities satisfy

93 2.5. A GENERAL SPIN-VECTOR DRIFT-DIFFUSION MODEL 93 t N c div x (D 1 ( x N c + x V N c )) = t Ns div x (D 1 ( x N s + x V N s ) + 2(D k 2 N s ) k=1,2,3 ) = Ω N s + (D 4 trd 4 )( N s ) 2 N s τ sf. (2.5.6) Here Ω = div x D 2 D 3 ( x V ) + H e, H e (x) = Ωe (x, v)m(v)dv R 3 (2.5.7) and the matrices D 1, D 2, D 3 and D 4 are given by (2.5.1). For two matrices A and B, A B is the product of A and B and div x (A) = x A = ( k ka ki ) i ; D k 2 is the k th row of D 2. We use x N s to denote the transpose of the Jacobian matrix of N s : x N s = ( xi N j s ) ij. Theorem (Maximum principle). Let N in L 2 (R 3, H + 2 (C)) be given and under the same hypothesis as for the last theorem, there exists a unique weak solution N(t, x) = N c(t, x) I 2 +N s (t, x). σ C ([, T ], L 2 (R 3, H 2 (C)) for any T > of (2.5.6) 2 with N(, x) = N in (x). In addition, for all t and x R 3, N(t,x) is an Hermitian and positive matrix (N(t, x) H + 2 (C)). Remark The right hand side of the limit equation (2.5.6) is the sum of a rotational term around a certain field Ω (2.5.7) and a relaxation terms arising from the spin-flip and non spin-flip scattering operators ( D 4 tr(d 4 ) is a negative matrix since D 4 is a symmetric positive definite matrix). The limiting effective field (2.5.7) contains an averaging of the even part Ω e and keeps traces via the matrices D 2 and D 3 from the odd part Ω o of the effective field in the kinetic equation. Before beginning the proof of these theorems, we have to introduce the four matrices D 1, D 2, D 3 and D 4 appearing in the limit model (2.5.6). These matrices keep traces from the collision operator and the spin-orbit interactions considered. This is the aim of the two following propositions. Proposition There exist a unique θ 1 (L 2 M )3 and θ 2 (L 2 M )3 such that Q(θ 1 I 2 ) = vm(v)i 2, θ 1 (v)dv =, (2.5.8) R 3 Q(θ 2 I 2 ) = Ω o (v)m(v)i 2, θ 2 (v)dv =, (2.5.9) R 3 where I 2 is the 2 2 identity matrix.

94 94 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS Proof. Using the properties of the collision operator introduced in Proposition ( ) and since vm(v)i 2 =, there exists ϑ 1 (L 2 M R 3 )3 such that Q(ϑ 1 ) = vm(v)i 2. The uniqueness of ϑ 1 is guarantied under the condition ϑ R 3 1 (v)dv =. It remains to prove that ϑ 1 is a scalar matrix. For this, we decompose ϑ 1 in the orthogonal basis {I 2, σ 1, σ 2, σ 3 } of the set of 2 2 hermitian matrices and we use the linearity of Q. Since Ω o is odd with respect to v, one can check similarly the existence of θ 2 satisfying (2.5.9). Proposition Let D 1, D 2, D 3 and D 4 be the 3 3 matrices defined respectively by D 1 = (θ 1 (v) v)dv, D 2 = (v θ 2 (v))dv, R 3 R 3 D 3 = ( Ω o (v) θ 1 (v))dv, D 4 = (θ 2 (v) (2.5.1) Ω o (v))dv R 3 R 3 where θ 1, θ 2 are given by (2.5.8), (2.5.9). The matrices D 1 and D 4 are symmetric positive definite and t D 3 = D 2. Proof. The components of D 1 verify D ij 1 = θ1(v).v i j dv = 1 θ1(v)i i 2 : v j M(v)I 2 dv R 2 3 R M 3 = 1 θ i j 1(v)I 2 : Q(θ1I 2 ) dv = 1 2 R M 2 θi 1I 2, Q(θ1I j 2 ) M. 3 Identically, one can calculate the components of D 2, D 3 and D 4 to find : D ij 2 = 1 2 θi 2I 2, Q(θ j 1I 2 ) M, D ij 3 = 1 2 θi 1I 2, Q(θ j 2I 2 ) M, D ij 4 = 1 2 θi 2I 2, Q(θ j 2I 2 ) M. The selfadjointness of Q provides that D 1 and D 4 are symmetric and that t D 3 = D 2. 3 To prove the positivity of D 1 (or D 4 ), let X R 3, and let fx 1 = X i θ1i i 2. Then, since fx 1 (KerQ), from (2.3.4) we have D 1 X, X = D ij 1 X i X j = 1 θ 2 1Id, i Q(θ1Id) j M X i X j i,j i,j = 1 2 i X i θ i 1Id, Q( j i=1 X j θ j 1Id) M = 1 2 f 1 X, Q(f 1 X) M α 1 2 f 1 X 2 M. Moreover, if X R 3 such that D 1 X, X = then, fx 1 =. This implies, by the 3 3 linearity of Q, that : X i Q(θ1I i 2 ) = and then X i v i M =. Finally, since i=1 (v i M) i is a family of linearly independent elements in L 2 M, we deduce that X =. Thus, D 1 (respectively D 4 ) is a symmetric positive definite matrix. i=1

95 2.5. A GENERAL SPIN-VECTOR DRIFT-DIFFUSION MODEL Diffusion limit : formal approach In this section, we will derive the model (2.5.6) by formally passing to the limit ε. Proposition If the solution of (2.5.2)-(2.1.2), F ε, has an Hilbert expansion with respect to ε in the form : F ε = F +εf 1 +O(ε), then F (t, x, v) = N(t, x)m(v) and the density matrix N satisfies (2.5.5). Proof. By inserting the expansion of F ε in (2.5.2) and comparing the terms corresponding to the same order of ε, we get Q(F ) =, Q(F 1 ) = (v x x V v )F i 2 [ Ω o σ, F ]. (2.5.11a) (2.5.11b) Therefore, F = N(t, x)m(v) and F 1 = θ 1 ( x N + N x V ) + i 2 θ 2 [ σ, N], where θ 1, θ 2 are given by (2.5.8) and (2.5.9) respectively. Integrating equation (2.5.2) with respect to v yields t N ε + div x J ε = S ε + Q sf (N ε ), (2.5.12) where N ε = F ε dv, J ε = 1 vf ε (t, x, v)dv and S ε = i [ Ω R ε 3 R 2 ε (x, v) σ, F ε (t, x, v)]dv. 3 R In addition, using the Hilbert expansion of F ε, one can calculate 3 formally the limit of each term of the last equation. Indeed, we have J ε = 1 vf ε (v)dv = 1 ε R ε ( vm(v)dv)n + vf 1 dv + O(ε) 3 R 3 R 3 = + vf 1 (v)dv + O(ε) = D 1 ( x N + N x V ) + i R 2 D 2([ σ, N]) + O(ε), 3 (2.5.13) and 2S ε = i [ Ω ε o σ, F ε ]+i [ Ω e σ, F ε ]dv = i [ Ω o σ, F 1 ]dv +i[h e σ, N]+O(ε) R 3 R 3 R 3 = i [ Ω o σ, θ 1 ( x N+N x V )]dv 1 [ Ω R 2 o (v) σ, θ 2 [ σ, N]]dv+i[H e σ, N]+O(ε). 3 R 3 Then, by a straightforward computation, one finds 2S ε = i[d 3 ( x + x V ) σ, N] i,j=1 D ij 4 [ e i σ, [ e j σ, N]] + i[h e σ, N] + O(ε), (2.5.14)

96 96 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS where { e 1, e 2, e 3 } is the euclidian basis of R 3. Let N = Nc 2 I 2 + N s σ, then with Lemma A.2.2 and the double cross product formula, a ( b c) = ( a c) b ( a b) c, one obtains 3 i,j=1 D ij 4 [ e i σ, [ e j σ, N]] = 4 = 4 3 i,j=1 3 i,j=1 D ij 4 e i ( e j N s ) σ D ij 4 ( N i s e j e i e j Ns ) σ, ( N i s = N s e i ) = 4(D 4 ( N s ) tr(d 4 ) N s ) σ. Replacing (2.5.13) and (2.5.14) in (2.5.12), passing to the limit ε, and using the fact that t D 2 = D 3 which implies that div x (D 2 [ σ, N]) = [(div(d 2 ) + D 3 ( x )) σ, N], one obtains (2.5.5) Diffusion limit : the rigorous approach This part is devoted to the proof of Theorem The first Lemma is a consequence of estimate (2.3.2). Lemma Let T > and let F ε C ([, T ]; L 2 M ) be the weak solution of (2.5.2). There exist F L 2 ([, T ], L 2 M ) and N L2 ([, T ] R 3, H 2 (C)) such that F ε F in L 2 t (L 2 M) weak and N ε N in L 2 t,x(h 2 (C)) weak. (2.5.15) In addition, we have N(t, x) = F (t, x, v)dv a.e. (t, x) R + R 3. R 3 Definition For all ε R +, we define the current J ε and the source spin-orbit term S ε by J ε (t, x) = 1 ε S ε (t, x) = i 2 R 3 vf ε (t, x, v)dv, (2.5.16) R 3 [ Ω ε (x, v) σ, F ε (t, x, v)]dv. (2.5.17) Lemma The current J ε and the term S ε given by (2.5.16), (2.5.17) are respectively bounded in L 2 ([, T ] R 3, (H 2 (C)) 3 ) and L 2 ([, T ] R 3, H 2 (C)) with respect to ε. Proof. By (2.3.3), there exists R ε L 2 ([, T ], L 2 M ) such that F ε = N ε M + εr ε and R ε L 2 t (L 2 M ) C. (2.5.18)

97 2.5. A GENERAL SPIN-VECTOR DRIFT-DIFFUSION MODEL 97 The current is then equal to : J ε (t, x) = vr ε (t, x, v)dv, and for all (t, x) R 3 R + R 3, we have with the Cauchy-Schwartz inequality T T ( ) 2 J ε (t, x) 2 (H 2 (C)) 3dtdx v R ε 2 dv dtdx R 3 R 3 R 3 ) R ε 2 L 2 t (L2 M (R ) v 2 Mdv. 3 Then, with (2.5.18), J ε is bounded in L 2 t,x((h 2 (C)) 3 ). By proceeding analogously, we obtain the boundedness of S ε in L 2 t,x(h 2 (C)). Proof of Theorem : As a consequence of Lemma 2.5.1, there exist J L 2 ([, T ] R 3, (H 2 (C)) 3 ) and S L 2 ([, T ] R 3, H 2 (C)) such that J ε J in L 2 t,x((h 2 (C)) 3 ) weak and S ε S in L 2 t,x(h 2 (C)) weak. If we pass formally to the limit in the equation (2.5.12) we get the continuity equation t N + div x J = S + Q sf (N). (2.5.19) In order to complete the limit equation (2.5.19), we have to find the relation between J, S and N. Indeed, multiplying equation (2.5.2) with θ1 I 2 M respect to v yields : J ε = R3 (v x F ε x V v F ε +ε t F ε ) θ1 M dv+iε 2 and integrating with R3 [ Ω ε σ, F ε ] θ1 R3 M dv+ε Q sf (F ε ) θ1 M dv. By passing to the limit, ε, we get J = v ( x N + x V N)θ 1 dv + i [ Ω R 2 o σ, N]θ 1 dv 3 R 3 = D 1 ( x N + x V N) + i 2 D 2[ σ, N]. (2.5.2) To find the relation between S and N, we apply the operation : i 2 (2.5.2). This yields [θ 2 σ,.] R M 3 dv on S ε i 2 [ Ω e σ, F ε ]dv = i [θ 2 σ, ε t F ε + v x F ε x V v F ε ] dv R 2 3 R M 3 ε [θ 2 σ, [ Ω 4 ε σ, F ε ]] dv R M + iε [θ 2 σ, Q sf (F ε )] dv 2 3 R M. 3

98 98 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS Taking ε goes to zero and using D 3 = t D 2, the last equation becomes (see the proof of Proposition for calculation details) S = i [θ 2 σ, v ( x N + x V N)]dv 1 [θ 2 σ, [ Ω 2 R 4 o σ, N]]dv + i 3 R 2 [H e σ, N] 3 = i 2 [D 3( x + x V ) σ, N] + (D 4 tr(d 4 ))( N s ) σ + i 2 [H e σ, N]. (2.5.21) For rigorous analysis, we have to use the weak formulation of (2.5.2) with different test functions. Remark first that (2.5.3) is also verified for test functions lie in the following space T = {ψ(t, x, v) C 1 ([, T ) R 6, H 2 (C)) compactly supported with respect to t and x and ψ and all its derivatives are polynomially increasing with respect to v i.e : n N, C R + / ψ(t, x, v) 2 + s ψ 2 C(1 + v ) n }. (2.5.22) s {t,x i,v i } In particular, if we take ψ = φ(t, x) C 1 c ([, T ) R 3, H 2 (C)) in (2.5.3), we obtain T T N ε, t φ 2 dtdx 1 F ε, v. x φ 2 dtdxdv = R ε 3 R 6 i T [ Ω R 2 ε (v). σ, F ε ]dv, φ 2 dtdx + Q sf (F ε ), φ 2 dtdxdv 3 R 3 R 3 + F in, φ(, x) 2 dxdv. (2.5.23) R 6 T This is nothing else but the weak formulation of the continuity equation (2.5.12) with initial condition N ε (, x) = F in (x, v)dv. (2.5.24) R 3 Passing to the limit ε in (2.5.23), one finds the limit continuity equation (2.5.19) in the distribution sense. It remains now to rigorously rely the current J and the term S with the density N. For this, one needs the following lemma which can be proved as Lemma Lemma Let θ 1 and θ 2 be given by (2.5.8), (2.5.9). Then, under Assumption and Assumption 2.5.1, we have θ 1 M C(1 + v ), 3 i=1 vi θ 1 M C(1 + v 2 ), (2.5.25)

99 2.5. A GENERAL SPIN-VECTOR DRIFT-DIFFUSION MODEL 99 θ 2 M + 3 i=1 xi θ 2 M C(1 + v )m, 3 i=1 where C stands for a generic nonnegative constant. vi θ 2 M C(1 + v )m+1, (2.5.26) This lemma shows that for all φ C 1 c ([, T ) R 3, H 2 (C)) each component of the vectorial function ψ = φ(t, x) θ 1 belongs to T. Using it as a test function in the M weak formulation (2.5.3), we get T T J ε, φ(t, x) 2 dtdx = ε F ε θ T 1, t φ 2 R 3 R 6 M dtdxdv+ F ε θ 1, v x φ v x V φ 2 R 6 M T F ε x V v θ T 1, φ 2 R 6 M dtdxdv + ε i R 6 2 [ Ω ε (x, v) σ, F ε θ 1 ], φ 2 M dtdxdv T + ε Q sf (F ε θ 1 ), φ(t, x) 2 R 6 M dtdxdv + ε θ 1 F in, φ(, x) 2 dxdv. (2.5.27) R 6 M Lemma Let Ω be a general vector field ( Ω = Ω o or Ω e ), then [ Ω σ, F ε ] converges weakly to [ Ω σ, N]M in L 2 ([, T ], L 2 M ). Proof. For all ψ L 2 ([, T ], L 2 M ), we have T [ Ω σ, F ε ], ψ 2 R M 6 T dtdxdv = F ε, [ Ω σ, ψ] 2 R M 6 dtdxdv ε T T N, [ Ω σ, ψ] 2 dtdxdv = [ Ω σ, N], ψ 2 dtdxdv. R 6 R 6 Using this lemma and with (2.5.25), it is simply to verify that we can pass to the limit in all the terms of equation (2.5.3). We obtain at the limit T T J, φ(t, x) 2 dtdx = N, ( x φ x V φ) v 2 θ 1 dtdxdv R 3 R 6 T + i R 2 [ Ω o (x, v) σ, N], φ 2 θ 1 dtdxdv 6 T T = N, D 1 ( x φ x V φ) 2 dtdx + i R 3 R 2 [D 2 σ, N], φ 2 dtdx. 3 This is the weak formulation of the current (2.5.2). Finally, to find weakly the relation between S and N given by (2.5.21), we choose now ψ = i[θ 2 σ, φ(t, x)] 2M for an arbitrary φ C 1 c ([, T ) R 3, H 2 (C)) as a test function in (2.5.3). In view of

100 1 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS (2.5.26) and Assumption 2.5.1, this is an admissible test function (i.e belongs to T ). One has + T T T ε 2 T S ε, φ(t, x) 2 dtdx i R 3 R 6 2 [ Ω e σ, F ε ], φ 2 dtdxdv = ε F ε, i R 6 2 [(v dtdxdv T x v x V )θ 2 σ, φ] 2 M F ε, i R 6 2 [ dtdxdv xv v (θ 2 σ), φ] 2 M ε T 4 T (Q sf F ε dtdxdv ), i[θ 2 σ, φ] 2 R 6 M ε 2 T F ε, i R 6 2 [θ dtdxdv 2 σ, t φ] 2 M F ε, i R 6 2 [θ dtdxdv 2 σ, v x φ] 2 M i[ Ω ε (x, v). σ, F ε dtdxdv ], i[θ 2 σ, φ] 2 R 6 M R 6 F in, i[θ 2 σ, φ(, x)] 2 dxdv M, (2.5.28) where, to obtain the left hand side of this equation, we have used the self adjointness of Q and the following identity. Lemma For each A, B and C in M 2 (C), we have A, [B, C] 2 = C, [A, B] 2. (2.5.29) One verifies easily that we can pass to the limit at all the terms of (2.5.28) to obtain T T T S, φ(t, x) 2 dtdx i R 3 R 3 2 [H e σ, N], φ 2 dtdx = N, i T R 6 2 [(v x v x V )θ 2 σ, φ] 2 dtdxdv N, i R 6 2 [θ 2 σ, v x φ] 2 dtdxdv 1 T [ Ω 4 o (x, v) σ, N], [θ 2 σ, φ] 2 dtdxdv. R 6 This can be rewritten, using identity (2.5.29) and the selfadjointness of all our matrices, as follows T T S, φ(t, x) 2 dtdx = i R 3 R 6 2 [(v x v x V )θ 2 σ, N], φ 2 dtdxdv T + i R 6 2 [θ 2. σ, N], v x φ 2 dtdxdv 1 T [θ 2 σ, [ Ω 4 o σ, N]], φ 2 dtdxdv R 3 T + i R 3 2 [H e σ, N], φ 2. This is the weak formulation of equation (2.5.21). The proof of Theorem is achieved Maximum Principle (Proof of Theorem 2.5.3) The existence of weak solution of (2.5.5) can be readily verified using semi-groupe technics and the fact that D 1 and D 4 are two symmetric definite positive matrices.

101 2.5. A GENERAL SPIN-VECTOR DRIFT-DIFFUSION MODEL 11 Let us just show that, for all (t, x), N(t, x) := N c(t, x) I 2 + N 2 s (t, x) σ is a non negative matrix. It is sufficient to verify that N c 2 N s since the eigenvalues of N are N c 2 ± N s. All the following computations can be made rigourously using the weak form of (2.5.6). Taking the scalar product of the second equation of (2.5.6) with N s, we get N s t ( N s ) div x (D 1 ( x N s + x V N s )) N s = 2(D 3 ( x ) N s ) N s + (D 4 trd 4 )( N s ) N s 2 N s 2. (2.5.3) τ sf Lemma We have div x (D 1 ( x N s + x V N s )) N s = N s div x (D 1 ( x N s + x V N s )) Proof. We have D 1 ( x N s ) : ( x N s ) + x N s D 1 ( x N s ). div x (D 1 ( x ) N s 2 ) = div x (D 1 ( x )( N s N s )) = i i ( j D ij 1 j ( N s N s )) = 2 i i ( j D ij 1 ( jns N s )) = 2 i (D ij 1 jn k s N k s ) i,j,k = 2 i,j,k i (D ij 1 jn k s ) N s k + 2 D ij 1 jn k s in k s i,j,k = 2 i,k i (D 1 ( x N s )) ik N k s + 2 i,k (D 1 ( x N s )) ik i N k s = 2div x (D 1 ( x N s )) N s + 2D 1 ( x N s ) : ( x N s ). In other side, we have div x (D 1 ( x N s 2 )) = 2div x ( N s D 1 ( x N s )) = 2 x N s D 1 ( x N s ) + 2 N s div x (D 1 ( x N s )). Identifying these two equations, one obtains div x (D 1 ( x N s )) N s = N s div x (D 1 ( x N s )) + x N s D 1 ( x N s ) D 1 ( x N s ) : ( x N s ). A similar calculations give div x (D 1 ( x V N s )). N s = N s div x (D 1 ( x V ) N s ).

102 12 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS Therefore, equation (2.5.3) becomes N { s t N s div x (D 1 ( x N s + x V N } s )) = D 1 ( x N s ) : ( x N s ) + 2(D 3 ( x ) N s ) N s + (D 4 trd 4 )( N s ) N s + x N s D 1 ( x N s ) 2 N s 2 Lemma We have, D 1 ( x N s ) : ( x N s ) + 2(D 3 ( x ) N s ) N s + (D 4 trd 4 )( N s ) N s + x N s D 1 ( x N s ). (2.5.31) τ sf D 1 ( x N s ) : ( x N s ) + 2(D 3 ( x ) N s ) N s + (D 4 trd 4 )( N s ) N s + x N s D 1 ( x N s ). (2.5.32) Proof. Let W (L 2 M (R3 )) 3 be the solution of Q(W ) = t ( x N s )(vm) + ( Ω o N s )M. We have W = t ( x N s )(θ 1 ) θ 2 N s. The operator Q is negative on L 2 M. Then, Indeed, for all ξ R 3, we have ( ) Q(W ) W R M dv (ξ) ξ = 3 Q(W ) W R M 3 (Q(W ) ξ)(w ξ) R M 3 This implies that, for all ξ R 3, ( Q(W ) W Q(W ξ), W ξ M tr R M 3 Particularly, taking ξ = N s, one gets the following inequality : dv is a negative matrix. dv = Q(W ξ), W ξ M. ) ( ) dv ξ 2 Q(W ) W = R M dv ξ 2. 3 N s 2 Q(W ) W R M dv (Q(W ) N s )(W N s ) dv, (2.5.33) 3 R M 3 which yields (2.5.32). Indeed, we have Q(W ) W M = t ( x N s )(v) t( x N s )(θ 1 ) t ( x N s )(v) (θ 2 N s ) t ( x N s )(θ 1 ) ( Ω o N s ) ( Ω o N s ) (θ 2 N s ) = ( x N s ) t( x N s ) : (v θ 1 ) ( x N s ) : (v (θ 2 N s )) ( x N s ) : (θ 1 ( Ω o N s )) ( Ω o N s ) (θ 2 N s ),

103 2.5. A GENERAL SPIN-VECTOR DRIFT-DIFFUSION MODEL 13 where we have used the following identity : A(v) B(w) = ( t A B) : (v w). Integrating with respect to v, the first term of the right hand side of the last equation is D 1 ( x N s ) : ( x N s ). In addition, ( x N s ) : v (θ 2 N s )dv = in j s v i (θ 2 N R 3 i,j R s ) j dv 3 = ( ) i N s j v i (θ 2 N s ) j dv N s j i (v i θ 2 N s ) j dv i,j R 3 i,j R 3 = ) i v i (θ 2 N j (R s ) N s dv N s j (v i i θ 2 N s ) j dv N s j (v i θ 2 ins ) j dv 3 i,j R 3 i,j R 3 = (( v i θ 2 i ) N s ) jn j s = ( t D 2 ( x ) N s ) jn j s = (D 3 ( x ) N s ) N s. j i R 3 j Similarly, one can verify that ( x N s ) : θ 1 ( Ω o N s )dv = (D 3 ( x ) N s ) N s. R Moreover, 3 ( Ω o N s ) (θ 2 N s )dv = ((θ 2 N s ) Ω o ) N s dv R 3 R ( 3 ) = ( Ωo θ 2 dv) N s ( Ω o N s )θ 2 N s = (tr(d 4 ) D 4 )( N s ) N s. R 3 R 3 Finally, a straightforward computations of the right hand side of (2.5.33) yield : Q(W ) N s = N s v x ( N s )M and W N s = N s θ 1 x ( N s ). Therefore, (Q(W ) N s )(W N s ) R M 3 = N s 2 R 3 v x ( N s ) θ 1 x ( N s )dv = N s 2 D 1 ( x N s ) x N s. All these computations together with inequality (2.5.33) give (2.5.32). To complete the proof of Theorem 2.5.3, (2.5.31) and (2.5.32) and the first equation of (2.5.6) imply that (N c 2 N s ) verifies t (N c 2 N s ) div x (D 1 ( x (N c 2 N s ) + x V (N c 2 N s ))). Moreover, since N c(, x) I 2 + N 2 s (, x) σ = N in (x) H 2 + (C) for all x R 3, then N c (,.) 2 N s (,.) and we conclude by the maximum principle satisfying by the scalar drift diffusion equation.

104 14 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS 2.6 SHE model Following the same strategy presented in the previous sections, other macroscopic models of two-component or spin vector types can be derived. In this section we are interested in the derivation of the so called SHE model. The Spherical Harmonic Expansion (SHE) model represents a diffusion approximation of the Boltzmann equation when the elastic collisions are supposed to be the dominant scattering mechanism. We refer to [2, 19] and references therein for details about the rigorous derivation of this model and the link with other macroscopic models (Drift-Diffusion, Energy-Transport, etc). The elastic collision operator takes the following form : Q el (F )(v) = σ(v, v )δ(e(v ) E(v))(F (v ) F (v))dv R 3 (2.6.1) where, for v = (E(v), ω(v)) R 3, E(v) = v 2 2 is the energy and ω(v) is the velocity angle. This operator makes relax the distribution function towards a function depending only on the energy of the particles (see Proposition 2.6.1). In other terms, the SHE model describes a situation following the relaxation of the momentum (ω) and preceding the energy relaxation of the particles. It describes an intermediate situation between the kinetic and the diffusion (drift-diffusion) descriptions. Before listing the properties of the elastic operator, we need to recall the Corea formula. For any ψ C (R 3 ) and E C 1 (R 3, R), the Corea formula says R 3 ψ(v)dv = + ( ) ψ(v)dn e (v) de (2.6.2) S e where, for any e R, S e = {v R 3, such that E(v) = e} and dn e (v) = ds e(v) E(v) with ds e (v) denotes the Euclidean surface element on S e. We will also denote, for every e R, N(e) := dn e (v). S e The following proposition summarizes the main properties of the elastic operator (2.6.1). We refer to [2] for the proof. Proposition The operator Q el (2.6.1) satisfies the following properties. 1. Q is a self-adjoint non negative operator on L 2 (R 3, H 2 (C)). 2. The nul set of Q el is given by N(Q el ) = { F L 2 (R 3, H 2 (C)) / F (v) = G(E(v)) for some G L 2 N(R) } with L 2 N(R) = {F (E) H 2 (C)/ R F (E) 2 2N(E)dE < + }. (2.6.3)

105 2.6. SHE MODEL Q el is coercive on N(Q el ) and the following coercivity inequality holds for C > Q el (F ), F L 2 C F P(F ) L 2, where P(F )(E) = F (v)dn E (v) is the orthogonal projection on N(Q el ). S E { } 4. Im(Q el ) = N(Q el ) = F L 2 (R 3, H 2 (C)) / F (v)dn E (v) = a.e.e R +. S E As for the previous section, the starting point will be the following scaled Boltzmann equation F ε t + 1 ε (v xf ε x V v F ε ) = 1 ε 2 Q el(f ε )+ i 2 [ Ω ε (x, v) σ, F ε ]+Q sf (F ε ), (2.6.4) where Ω ε is given by (2.5.1) and Q el by (2.6.1). Using the properties of Q el, one can redefine the matrices D i (2.5.1) as follows. Lemma Let χ 1, χ 2 (L 2 (R 3 )) 3 be the unique solutions of Q el (χ 1 ) = v, χ 1 dn E (v) = a.e. E R +, S E Q el (χ 2 ) = Ω o, χ 2 dn E (v) = a.e. E R +, S E and let D 1, D 2, D 3 and D 4 be the following matrices D 1 (E) = (χ 1 v)dn E (v), D 2 (x, E) = (v χ 2 )dn E (v), S E S E D 3 (x, E) = ( Ω o χ 1 )dn E (v), D 4 (x, E) = (χ 2 Ω o )dn E (v). S E S E (2.6.5) (2.6.6) Then, for almost x R 3 and E R +, D 1 (E) and D 4 (x, E) are two symmetric positive definite matrices and t D 2 = D 3. In addition we have, D 1 (E) C (v v)dn E (v), D 4 (x, E) C ( Ω o Ω o )dn E (v), (2.6.7) S E S E with a constant C > independent on x and E. The proof of this lemma is straightforward, one can see [2] for details. Taking an Hilbert expansion of F ε around ε = F ε = F + εf 1 + ε 2 F , inserting it in (2.6.4) and identifying equal powers of ε, leads to the equations Q el (F ) =, (2.6.8)

106 16 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS Q el (F 1 ) = v. x F x V. v F i 2 [ Ω o σ, F ], (2.6.9) Q el (F 2 ) = t F + v. x F 1 x V. v F 1 i 2 [ Ω o σ, F 1 ] i [ ] Ωe σ, F Q sf (F ). 2 (2.6.1) The first equation implies the existence of an energy dependent function F(t, x, E) such that F (t, x, v) = F(t, x, E(v)). The right hand side of the second equation (2.6.9) becomes v. x F x V. v F i 2 [ Ω o σ, F ] = v ( x F x V E F) i 2 [ Ω o σ, F] and with (2.6.5), we have F 1 = χ 1 ( x F x V E F) + i 2 [χ 2 σ, F]. (2.6.11) Using the properties of the Q el (Proposition 2.6.1), equation (2.6.1) admits a solution if and only if the averaging of its right hand side over S E is equal to zero for almost every E R. This gives after integration over S E N(E) t F + x.j x V. E J = S + i [ ] Ωe (x, v)dn E (v) σ, F +Q sf (F) (2.6.12) 2 S E with J(t, x, E) = vf 1 dn E (v) (2.6.13) S E and S = i [ Ω 2 o σ, F 1 ]dn E (v). (2.6.14) S E To obtain the third term of (2.6.12), we proceed as follows. We write for any ψ = ψ(e) C (R), a continuous compactly supported function, using the Corea formula (2.6.2), v F 1 ψ(e(v))dv = F 1 v ψ(e(v))dv = vf 1 ψ (E)dN E de R 3 R 3 R S E = ( vf 1 dn E )ψ (E)dE = Jψ (E)dE. R S E R One deduces that v F 1 dn E = E J. Moreover, an analogous computations as in S E the proof of Proposition lead and J = D 1 ( x F x V E F) + i 2 [D 2( σ), F] S = i 2 [D 3( x x V E ) σ, F] + (D 4 tr(d 4 ))( F s ) σ with F s σ = F tr(f) I 2. Thus, to summarize one can deduce the following theorem. 2

107 2.7. OTHER FLUID MODELS FOR SEMICONDUCTOR SPINTRONICS 17 Theorem Let T >, F in L 2 ([, T ] R 6 ), then under Assumptions 2.2.1, 2.2.2, 2.5.1, the weak solutions F ε of (2.6.4)-(2.1.2) converges weakly to F in L ([, T ], L 2 (R 6, H 2 (C))) such that F (t, x, v) = F(t, x, v 2 ) for some F(t, x, E) 2 L ([, T ], L 2 N (dxde)). In addition F satisfies the following SHE model with spin precession and relaxation effects : N(E) t F + ( x x V E ) J = i 2 [ Ω SHE σ, F] + (D 4 tr(d 4 ))( F s σ) + Q sf (F), with J = D 1 ( x F x V E F) + i[d 2 ( σ), F] Ω SHE (x, E) = ( x x V E ) D 2 (x, E) + Ωe (x, v)dn E (v) S E and where F s σ = F tr(f) I 2 is the spin part of F Other fluid models for semiconductor spintronics Review of macroscopic models and moment method. In microelectronics, the macroscopic or fluid models describe the evolution of macroscopic averaged quantities of the distribution function f. These quantities are usually the particle number density n(t, x), the current density n(t, x)u(t, x) (where u is the mean velocity) and the energy density W(t, x). They are called the moments of f and are given by n nu W = R 3 1 v v 2 2 f(t, x, v)dv. (2.7.1) These quantities evolve according to balance equations such as mass, momentum, and energy balance. The macroscopic models (Euler, ET, SHE, Drift-Diffusion) can be obtained from the kinetic equation t f + v x f x V v f = Q(f) (2.7.2) according to the conservations of the collision operator by using the moment method [1, 31, 6, 26, 28, 29, 45]. As an example, assume that the collision operator admits the following conservation properties Q(f) R 3 1 v v 2 2 dv =. (2.7.3)

108 18 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS The moment method consists in multiplying the Boltzmann equation (2.7.2) by 1 and integrating it with respect to v. This gives with (2.7.1) and (2.7.3) v v 2 2 t n nu + x f(t, x, v) R W 3 1 v v 2 2 vdv = n x V nu x V. It can be written also as t n + x (nu) = t (nu) + x Π = n x V t W + x Φ = nu x V, (2.7.4) where Π(t, x) and Φ(t, x) are respectively the pressure tensor and the energy flux given by R3 Π = f(v v)dv and Φ = f v 2 R 2 vdv. 3 System (2.7.4) is not closed because Π and φ can not be expressed in terms of n, nu, W. To close system (2.7.4), one needs to find a distribution function which allows to compute these quantities in terms of the conserved variables (2.7.1). The Levermore s methodology [31] consists in using the entropy minimization principle. Let n, T R + and u R 3 be fixed, the entropy minimization problem is stated as 1 n min f h(f)/ f dv = nu R, (2.7.5) 3 W v v 2 2 with W = n u nt. The entropy h is a convexe function. It describes the statistics 2 2 of the particles. In the Boltzmann statistics, h is given by h(x) = x(ln x 1). (2.7.6) Problem (2.7.5) with (2.7.6) admits a unique solution given by the Maxwellian distribution M n,u,t (v) = n (2πT ) 3 2 ) v u 2 exp (. (2.7.7) 2T The parameters n, u and T are such that n, nu and W respectively are its density, momentum and energy R 3 M n,u,t 1 v v 2 2 dv = n nu W.

109 2.7. OTHER FLUID MODELS FOR SEMICONDUCTOR SPINTRONICS 19 Replacing f by M n,u,t in (2.7.4), one obtains the Euler model t n + x (nu) =, t (nu) + x (nu u) + x (nt ) = n x V, t W + x (Wu) + x (nut ) = nu x V. The Euler model can be obtained from the Boltzmann equation by hydrodynamic limit. It consists in taking the scaled equation t f ε + v x f ε x V v f ε = 1 ε Q(f ε ) and getting ε goes to zero. In the hydrodynamic limit, we don t need to know the exact form of the collision operator. It is sufficient to fix some properties satisfied by Q. To obtain for example the Euler model, Q must satisfies the conservation properties (2.7.3) ; the null space (or the local equilibrium) must be given by the maxwellians (2.7.7). In addition, Q must satisfies some entropy decay. In our case, the distribution is a matrix valued function. To derive maxwellian functions, we propose to study the following entropy problem. Let M = (M i ) d i=1 be a vector of 2 2 hermitian matrices, M i. It represents the vector of the moments of the distribution function. Let µ(v) = (µ i ) d i=1 be the vector of monomials of v (1, v, v 2,...). The entropy minimization problem writes as 2 min F {H(F ) = tr(h(f ))/ R 3 µ(v)f dv = M}. (2.7.8) The entropy writes as H(F ) = tr(h(f )) with h is a convexe function. The expression h(f ) refers to the matrix obtained by acting the function h onto F by functional calculus i.e. h(f ) has the same eigenbasis as F and has eigenvalues h(λ s ) where λ s are the eigenvalues of F. The solution of (2.7.8) is easily found and is given by M α = (h ) 1 ( α µ(v)) if the derivative of h is invertible. The vector α = (α i ) d i=1 is a vector of 2 2 hermitian matrices such that R 3 M α µ i (v)dv = M i, i d. (2.7.9) When h is given by the Boltzmann statistics (2.7.6), the maxwellian becomes M α = exp( α µ). (2.7.1)

110 11 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS Energy-Transport model The Energy-Transport (ET) model is a diffusion model constituted of a balance equation for the electron density and an energy balance equation [3, 4, 14, 19]. To derive ET Model for semiconductor spintronics, we take the Boltzmann equation in the diffusion scaling (2.5.2). The collision operator Q must now conserve only the mass and the energy. This means R 3 Q(f) ( 1 v 2 2 ) dv =. (2.7.11) Following the Levermore s moment method, for any distribution function F (v), we construct the maxwellian( M F given by (2.7.6), µ(v) = the solution of { min G 1 v 2 2 H(G) = G, (log(g) I 2 ) 2 / associated ) to F as the solution of (2.7.8) with h, and M = R 3 F µ(v)dv. More precisely, M F is R 3 (G F ) ( 1 v 2 2 ) dv = }. Then, by (2.7.1)-(2.7.9), M F is given by M F (v) = M[A, C] := exp ) (A + C v 2 2 (2.7.12) with A H 2 (C) and C H 2 (C) are such that ( ) M[A, C] dv = R 3 1 v 2 2 R 3 F ( 1 v 2 2 ) dv. (2.7.13) Let now the collision operator be given by the following BGK simple form Q(F ) = M[A, C] F, (2.7.14) where M[A, C] is defined by (2.7.12) and A and C are relied to F with (2.7.13). Let us, before listing the properties of this operator, gives some properties satisfied by log as function on H 2 + (C). Lemma For all F H 2 + (C), log(f ) is the unique matrix in H 2 (C) satisfying e log(f ) = F. Writing F = f c I 2 + f s σ, then we have ( log(f ) = 1 2 log(f c 2 f s 2 )I log f c f ) s fs f c + f s f σ (2.7.15) s if f s. Its eigenvalues are then log(f c f s ) and log(f c + f s ).

111 2.7. OTHER FLUID MODELS FOR SEMICONDUCTOR SPINTRONICS For all F, G H 2 + (C), the following properties hold. (a) If F and G commutate, then log(f G) = log(f ) + log(g) (b) If F is invertible then log(f 1 ) = log(f ) (c) The function log is strictly increasing which means that and (F G), (log F log G) 2 (2.7.16) (F G), (log F log G) 2 = F = G. (2.7.17) The collision operator (2.7.14) satisfies the following properties. (i) By (2.7.13), Q preserves the mass and the energy i.e. (2.7.11) is satisfied. (ii) Its null space is spanned by the maxwellians. More precisely Q(F ) = (A, C) = (A(t, x), C(t, x)) s.t. F = M[A, C] = exp (iii) By construction of M F decay ) (A + C v 2. 2 and (2.7.16)-(2.7.17), we have the following entropy R 3 Q(F ), log F 2 dv, (2.7.18) with equality if and only if F = M[A, C]. Indeed, using (2.7.13), we have Q(F ), log(m[a, C]) 2 dv = R 3 Therefore, with (2.7.16), we have R 3 M[A, C] F, (A + C v 2 2 ) dv =. 2 Q(F ), log F 2 dv = Q(F ), log F log(m[a, C]) 2 dv R 3 R 3 = M[A, C] F, log F log(m[a, C]) 2 dv. R 3 We state now the main result of this subsection which consists on formal derivation of ET model with spin rotation and relaxation effects. Theorem Let F ε be the solution of (2.5.2)-(2.1.2) with Q given by (2.7.14). Then, formally, F ε F as ε, where

112 112 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS F (t, x, v) = M[A, C] = exp t R 3 M[A, C] + i 2 R 3 = i 2 ( R 3 R 3 1 v 2 2 ) dv ) (A(t, x) + C(t, x) v 2 and (A, C) are solutions of 2 ( T 2 M[A, C] i 2 T [ Ωo σ, M[A, C] ] ) ( 1 [ Ω o σ, T M[A, C] i 2 [ Ω o σ, M[A, C]] [ ] ( Ωe 1 σ, M[A, C] v 2 2 v 2 2 ] ( 1 ) ( dv + Q sf (M[A, C]) R 3 where T denotes the transport operator : T = v x x V v. ) v v 2 2 dv ) ) dv dv, (2.7.19) Let us write the Energy-Transport model (2.7.19) under the form of conservation laws. For any pair of functions (A(t, x), C(t, x)), we respectively denote by n[a, C], W[A, C], Π[A, C] and Q[A, C] the particle and energy densities, the pressure tensor and the heat flux tensor associated to A and C. They are given by ( ) ( ) n[a, C] = M[A, C] dv (2.7.2) W[A, C] R 3 and ( Π[A, C] Q[A, C] ) = R 3 ( v v v 2 2 v v ) 1 v 2 2 M[A, C]dv. (2.7.21) In addition, we will denote by Π Ωo [A, C] and Q Ωo [A, C] the following tensors associated to A, C and Ω o : ( ) Π Ωo [A, C] Q Ωo [A, C] ( = i R 3 1 v 2 2 ) [ (v Ω ] o )( σ), M[A, C] dv. (2.7.22) Lemma With notations (2.7.2), (2.7.21) and (2.7.22), the spin-vector Energy- Transport model (2.7.19) can be equivalently written t n + x J n = i [ ( ) ΩET σ, M[A, C]] dv + Ωo ( Ω 2 o M s [A, C])dv σ + Q sf (n) (2.7.23) R 3 R 3 t W + x J w + J n x V = i [ ΩET v 2 σ, M[A, C]] 2 2 dv + R 3 (R3 ) Ωo ( Ω o M s [A, C]) v 2 2 dv σ + Q sf (W). (2.7.24)

113 2.7. OTHER FLUID MODELS FOR SEMICONDUCTOR SPINTRONICS 113 The mass and energy fluxes are given by J n = ( x Π + n x V Π Ωo ) (2.7.25) J w = ( x Q + W x V + Π x V Q Ωo ) (2.7.26) where Π, Q, Π Ωo and Q Ωo are nonlinear functionals of n and W through (2.7.21), (2.7.22) and (2.7.2). Moreover, in the right hand side of the conservation equations (2.7.23)-(2.7.24), the effective field Ω ET is given by Ω ET = div x (v Ω o ) x V v Ωo + Ω e and M s [A, C] is the spin part of M[A, C], M s [A, C] = M[A, C] 1 2 tr (M[A, C]) I 2. Remark The model (2.7.23)-(2.7.24) is constituted of two continuity equations on the density and the energy coupled via the maxwellian M[A, C] with additional rotational and relaxation terms as in the drift diffusion case. The first term of the right side of (2.7.23) (or (2.7.24)) describes the precession of the spin vector part of M[A, C] around the effective field Ω ET. The other terms are relaxation terms. We discuss now an important property satisfied by the Energy-Transport model (2.7.23)-(2.7.24) : the entropy dissipation. This property is a direct consequence of our derivation, thanks to the entropy dissipation inequality satisfied by the collision operator (2.7.18). More precisely, we define the entropy by S(n, W) = M[A, C], log(m[a, C]) I 2 2 dxdv R 6 = M[A, C], A + C v 2 R 2 I 2 dxdv 6 2 = ( A, n 2 + C, W 2 tr(n))dx (2.7.27) R 3 where (A, C) and (n, W) are related through (2.7.2). The function S is a strictly convex functional of (n, W) and we have : Lemma Let (n, W) solves the Energy-Transport model given by the previous lemma. Then, the fluid entropy S(n, W) is a decreasing function of time d S(n, W). (2.7.28) dt Proof of Theorem We present now the formal derivation of (2.7.19). We assume that F ε F as ε in a space of smooth functions. Multiplying

114 114 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS (2.5.2) by ε 2 and letting ε, one obtains Q(F ) =. This implies that there exist (A(t, x), C(t, x)) such that F = exp(a + C v 2 ). Now, we introduce the following 2 (Chapman-Enskog) expansion F ε = M F ε + εf ε 1, (2.7.29) where M F ε is the maxwellian associated to F ε. Thus, we have Inserting this expression into (2.5.2), we get F ε 1 = 1 ε Q(F ε ). (2.7.3) F ε 1 = ε t F ε T F ε + iε 2 [ Ω ε σ, F ε ] + εq sf (F ε ). Therefore, as ε, F ε 1 F 1 such that Next, multiplying (2.5.2) by F 1 = T F + i 2 [ Ω o σ, F ]. (2.7.31) ( 1 v 2 2 ) and using the conservation properties (2.7.11), we get ( ) t F ε 1 v dv + 1 ( ) T F ε 1 2 v dv R ε R 3 2 = i ( ) ( ) [ Ω 2 ε σ, F ε 1 ] v dv + Q 2 sf (F ε 1 ) v dv 2 R 3 2 R 3 2 ( ) 1 Now, using (2.7.29) and the fact that T M F ε v dv = and 2 R 3 2 ( ) [ Ω 1 o σ, M F ε] v dv = since T M 2 F ε and Ω o are odd vectors with respect R 3 2 to v, one obtains ( ) ( ) t F ε 1 v dv + T F ε v dv = i ( ) [ Ω 2 R 3 2 R 2 o σ, F ε 1 1 ] v dv R i ( ) ( ) [ Ω 2 e σ, F ε ] dv + Q sf (F ε ) dv. R 3 R 3 This gives when ε ( ) ( ) 1 1 t F v dv + T F R 2 1 v dv = i ( ) [ R 2 1 Ω o σ, F 1 ] v dv R i ( ) ( ) [ Ω 2 e σ, F ] dv + Q sf (F ) dv. R 3 R 3 1 v v v v 2 2

115 2.7. OTHER FLUID MODELS FOR SEMICONDUCTOR SPINTRONICS 115 Finally, inserting the expressions of F and F 1 into the last equation leads to (2.7.19). Proof of Lemma We multiply (2.5.2) by log(f ε ), integrate with respect to x and v t F ε, log F ε 2 dxdv + 1 T F ε, log F ε 2 dxdv = 1 Q(F ε ), log F ε R ε 6 R ε dxdv R 6 + i R 2 [ Ω ε σ, F ε ], log F ε 2 dxdv + Q sf (F ε ), log F ε 2 dxdv. (2.7.32) 6 R 6 We have d dt F ε, log F ε I 2 2 dxdv = d tr(f ε (log F ε I 2 ))dxdv R dt 6 R 6 = d tr(h(f ε ))dxdv dt R 6 where h = x(logx 1). In other side we have d dt tr(h(f ε )) = δ F (trh(.)) F ε t F ε = tr(h (F ε ) t F ε ) where δ F (tr(h(.))) denotes the Gâteaux derivative of tr(h(.)) (see also [18] for more details). We deduce that d dt tr(h(f ε )) = tr(log F ε t F ε ) = t F ε, log F ε 2. Therefore, t F ε, log F ε 2 dxdv = d F ε, log F ε I 2 2 dxdv. R dt 6 R 6 A similar computations give T F ε, log F ε 2 dxdv = T ( F ε, log F ε I 2 2 )dxdv =. R 6 R 6 Moreover, we have i R 2 [ Ω ε σ, F ε ], log F ε 2 dxdv = ( Ω ε f s ε ) σ, log F ε 2 dxdv 6 R 6 = 2( Ω ε f ε s ) g ε s, where f ε s is the spin part of F ε and g ε s the spin part of log(f ε ). With (2.7.15), g ε s is parallel to f ε s and we conclude that R 6 i 2 [ Ω ε σ, F ε ], log F ε 2 dxdv =.

116 116 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS In other side, using the entropy decay property of Q (2.7.18), the first term of the right hand side of (2.7.32) is negative. In addition, the third term of the right hand side of (2.7.32) is also negative. Indeed, we have ( ) ( ) tr(f tr(f ε ) 2F ε ε ) tr(f ε ), log I 2 = 2 log tr( f 2 2 s ε σ) =. Then, Q sf (F ε ), log F ε 2 dxdv = 1 R τ 6 sf = 1 τ sf = 1 2 tr(f ε ) 2F ε, log F ε 2 dxdv R 6 ( ) tr(f tr(f ε ) 2F ε, log F ε ) log R 2 6 τ sf tr(f ε ) 2F ε, log(2f ε ) log(tr(f ε )) 2 dxdv R 6 (in view of (2.7.16)). As a conclusion, the solution of (2.5.2) with (2.7.14) satisfies d F ε, log F ε I 2 2 dxdv. dt R 6 To complete the proof, we pass to the limit ε. This ends the proof of the dxdv 2 lemma since F ε converges to F = exp(a + C v 2 ) where (A, C) is the solution of the 2 spin-vector Energy-Transport model (2.7.19) Drift-Diffusion with Fermi-Dirac statistics The nonlinear BGK approximation of the collision operator for Fermi-Dirac statistics, when we consider a Boltzmann equation with scalar distribution function f, takes the form Q(f) = σ(v, v ) [M(v)f(v )(1 f(v)) M(v )f(v)(1 f(v ))] dv R 3 (2.7.33) where M is given by (2.1.5). We refer to [27] for the study of such operator. We list in the following lemma the main properties satisfied by (2.7.33). Lemma Assume that the cross-section verifies Assumption Let A = {f L 1 (R 3 ) ; < f < 1 a.e.} be the set of admissible functions. Then, 1. the operator Q given by (2.7.33) is a bounded operator on A in L 1 (R 3 ). 2. For any f A, we have the following entropy inequality ( ) f H(f) = Q(f) log dv, (2.7.34) R (1 f)m 3 V

117 2.7. OTHER FLUID MODELS FOR SEMICONDUCTOR SPINTRONICS 117 Inspired from the scalar case, we consider the following space of admissible functions where M V. is the maxwellian associated to the potential V given by M V := 1 e 1 (2π) 3 2 v 2 V The following statements are equivalent : (i) f A and H(f) = (ii) f A and Q(f) = (iii) there exists µ R such that f(v) = (1 + e v 2 2 µ ) 1. A = {F L 1 (R 3 ; H 2 (C)) ; F (v) I 2 a.e.v} (2.7.35) when, for two hermitian matrices A and B, the relation A B means that (B A) is a positive defined matrix. We will construct a collision operator Q F D with 2 2 hermitian matrix value admitting the following properties inspired also from the scalar case : 1. Q F D preserves the mass : R 3 Q F D dv =. 2. For every F A, Q admits the following entropy inequality H(F ) = Q F D (F ) log(f (I 2 F ) 1 M 1 V )dv R 3 3. H(F ) = Q F D (F ) = F = M F. Here M F denotes the maxwellian associated to F with the Fermi-Dirac statistics given by ( ) 1 M F = I 2 + e v 2 2 Π (2.7.36) where Π is a unique matrix belonging to H 2 (C) such that M F dv = R 3 F dv. R 3 (2.7.37) We set as for the Energy-Transport case Q F D (F ) = M F F. (2.7.38) The maxwellian (2.7.36) with (2.7.37) can also be seen as the solution of an entropy minimization problem. When the only conserved moment is the density or the mass ; the convenient entropy concept is the relative entropy. It is given for Fermi-Dirac statistics by h F D (F ) = F ) (log F + v V + (I 2 F ) log (I 2 F ). (2.7.39)

118 118 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS For any F A fixed, the solution of min G A { tr(h F D (G))/ } Gdv = R 3 F dv R 3 is nothing but M F given by (2.7.36)-(2.7.37). Besides, one can verify, by direct computations using decomposition of the matrices into charge and spin parts, that problem (2.7.36)-(2.7.37) is well posed and admits a unique solution. This is the aim of the following lemma and proposition where their proofs are left to the reader. Lemma Let Π = π c I + π s σ H 2 (C) and F L 1 (R 3, H + 2 (C)), then M F defined by (2.7.36) belongs to A. In addition, we have M F = α c I 2 + α s σ (2.7.4) such that α c = F D(π c ), α s = if π s = and if π s, we have α c = 1 2 (F D(π c + π s ) + F D(π c π s )) (2.7.41) α s = 1 2 (F D(π c + π s ) F D(π c π s )) π s π s. (2.7.42) The function F D(.) on R denotes the Fermi-Dirac distribution given by F D(φ) = ( 1 + e v 2 2 φ ) 1, φ R. (2.7.43) Proposition Let F = F (v) L 1 (R 3, H + 2 (C)). Then, there exists a unique 2 2 hermitian matrix Π independent of v satisfying (2.7.37). In addition, if we write Π = π c I 2 + π s σ and ρ := F (v)dv = ρ c I 2 + ρ s σ, then (2.7.37) is equivalent to R 3 where F D is given by (2.7.43). R 3 F D(π c + π s )(v)dv = ρ c + ρ s F D(π c π s )(v)dv = ρ c ρ s R 3 π s and π s = ρ s if ρ s, ρ s (2.7.44) The newt proposition summarizes the main properties of the collision operator (2.7.38). Proposition The operator Q F D defined by (2.7.38), (2.7.36) and (2.7.37) satisfies the following properties.

119 2.7. OTHER FLUID MODELS FOR SEMICONDUCTOR SPINTRONICS Q F D is bounded on L 1 (R 3 ; H + 2 (C)) i.e. there exists a constant C such that for all F L 1 (R 3 ; H + 2 (C)) Q F D (F ) L 1 (R 3 ;H + 2 (C)) C F L 1 (R 3 ;H + 2 (C)). (2.7.45) 2. For all F A, we have the following H-Theorem (or entropy decay) H(F ) := Q F D (F ) : [ log(f (I 2 F ) 1 M 1 V )]. (2.7.46) R 3 3. The following statements are equivalent (i) F A and H(F ) =. (ii) F A and Q F D (F ) =. (iii) there exists Π H 2 (C) such that F = (I 2 + e v 2 2 Π ) 1. Proof. 1. For any Π H 2 (C), the matrix M F given by (2.7.4), (2.7.41), (2.7.42) satisfies M F 2 2 = (α c I 2 + α s σ) : (α c I 2 + α s σ) = 2(α 2 c + α s 2 ) = F D(π c + π s ) 2 + F D(π c π s ) 2 [F D(π c + π s ) + F D(π c π s )] 2. Let F L 1 (R 3 ; H + 2 (C)) and let Π be the solution of (2.7.37). Then, in view of (2.7.44), we have M F 2 dv F D(π c + π s )dv + F D(π c π s )dv R 3 R 3 R 3 ρ c = f c dv R 3 f c dv R 3 F 2 dv. R 3 Thus, Q F D (F ) 2 R 3 M F 2 dv + R 3 F 2 dv 2 R 3 F 2 dv. R 3 2. For all F A, it is simple to verify that F (I 2 F ) 1 belongs to H + 2 (C) and F commutate with (I 2 F ) 1. Then, log(f (I 2 F ) 1 ) is well defined and log(f (I 2 F ) 1 ) = log(f ) log(i 2 F ). Let now Π be the solution of (2.7.37). Then, we have M F (I 2 M F ) 1 M 1 V = (2π) 3 2 e Π+V. Indeed, M F (I 2 M F ) 1 = = ( ) 1 [ ] 1 I 2 + e v 2 2 Π (I 2 M F ) 1 = (I 2 M F )(I 2 + e v 2 2 Π ) ( I 2 + e v 2 2 Π I 2 ) 1 = e v 2 2 e Π = (2π) 3 2 MV e Π+V.

120 12 CHAPTER 2. HIERARCHY OF KINETIC AND MACROSCOPIC MODELS Therefore, log(m F (I 2 M F ) 1 M 1 V ) = 3 2 log(2π)i 2 + Π + V. Since Π, V are independent of v and Q F D (F )dv = then, R 3 H(F ) = Q F D (F ) : (log(f (I 2 F ) 1 M 1 V ) 3 R 2 log(2π)i 2 Π V ) 3 = Q F D (F ) : (log(f (I 2 F ) 1 M 1 V ) log(m F (I 2 M F ) 1 M 1 V )) R 3 [ ] = (M F F ) : R 3 log(f ) log(m F ) + log(i 2 M F ) log(i 2 F ). In view of (2.7.16), (M F F ) : (log(f ) log(m F )) and R 3 (M F F ) : (log(i 2 M F ) log(i 2 F )) R 3 = [(I 2 F ) (I 2 M F )] : (log(i 2 M F ) log(i 2 F )). R 3 We deduce that H(F ), F A. 3. These equivalences are obvious using (2.7.17). We state now the main results of this subsection where their proofs are similar to those presented in the last subsection. Theorem Let F ε be the solution of (2.5.2)-(2.1.2) with the collision operator Q F D given by (2.7.38). Then, formally, F ε F as ε, where F (t, x, v) = (I 2 + e v 2 2 Π(t,x) ) 1 and Π is solution of t n[π] + J = i 2 R 3 [ Ω F D σ, where the mass flux J is given by ( I 2 + e v 2 2 Π(t,x) ) 1 ] dv ( ) + Ωo ( Ω o F o s )dv σ + Q sf (n[π]), (2.7.47) R 3 J = D[Π] x V n[π] + D Ωo [Π]. (2.7.48) The density n[π], D[Π] and D Ωo [Π] are nonlinear functionals of Π given by n[π] = R 3 ( I 2 + e v 2 2 Π(t,x) ) 1 dv, (2.7.49)

121 REFERENCES 121 and ( ) 1 D[Π] = (v v) I 2 + e v 2 2 Π(t,x) dv, R 3 [ ( ) ] 1 D Ωo [Π] = i (v Ω o )( σ), I 2 + e v 2 2 Π(t,x) dv. R 3 The Fermi-Dirac effective field Ω F D is similar to the one obtained in the derivation of energy transport model. It is given by Ω F D = div x (v Ω o ) x V v Ωo + Ω e. Finally, the F s is the spin part of F = (I 2 + e v 2 2 Π(t,x) ) 1. The entropy function is now defined by, where F = R 3 F dv, S(n) = = = (I 2 + e v 2 2 Π ) 1 and n = F, log(f (I 2 F ) 1 M 1 V ) 2dxdv + tr(log(i 2 F ))dxdv R 6 R 6 F, 3 R 2 log(2π)i 2 + Π + V 2 dxdv + tr(log(i 2 F ))dxdv 6 R ( ) log(2π)tr(n) + n, Π + V 2 dx + tr(log(i 2 F ))dxdv. R 6 R 3 Proposition Let Π and n := n[π] solve the Drift-Diffusion system (2.7.47)- (2.7.48). Then, the Fermi-Dirac fluid entropy S satisfies d dt S(n) t V tr(n)dx. (2.7.5) R 3 If the potential V is independent of time, then S(n) is a decreasing function of time d dt S(n). References [1] C. Bardos, F. Golse, and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys., 63 (1991), pp [2] N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), pp [3] N. Ben Abdallah, P. Degond, and S. Génieys, An Energy-Transport model derived from the Boltzmann equation of semiconductors, J. Stat. Phys., 84 (1996), pp

122 122 REFERENCES [4] N. Ben Abdallah, S. Desvillettes, and S. Génieys, On the convergence of the Boltzmann for the semiconductors toward the energy transport model, J. Stat. Phys., 98 (2), pp [5] J. A. Bittencourt, Fundamentals of plasma physics, Pergamon Press, Oxford, [6] F. Bouchut, F. Golse, and M. Pulvirenti, Kinetic equations and asymptotic theory, B. perthame, L. Desvillettes Eds, Series in Appl. Math., bf 4, Gauthier-Villars (2). [7] A. Bournel, Magnéto-électronique dans des dispositifs à semiconducteurs, Ann. Phys. Fr, 25 (2), pp [8] A. Bournel, V. Delmouly, P. Dollfus, G. Tremblay, and P. Hesto, Theoretical and experimental considerations on the spin field effect transistor, Physica E, 1 (21), pp [9] A. Bournel, P. Dollfus, E. Cassan, and P. Hesto, Monte Carlo study of spin relaxation in AlGaAs/GaAs quantum wells, Applied Physics Letters, 77 (2), pp [1] Y. A. Bychkov and E. I. Rashba, Oscillatory effects and the magnetic susceptibility of carriers in inversion layers, Journal of Physics C : Solid State Physics, 17 (1984), pp [11] S. Datta and B. Das, Electronic analog of the electro-optic modulator, Applied Physics Letters, 56 (199), pp [12] P. Degond, Mathematical modelling of microelectronics semiconductor devices, in Some current topics on nonlinear conservation laws, vol. 15 of AMS/IP Stud. Adv. Math., Amer. Math. Soc., Providence, RI, 2, pp [13] P. Degond, G. Gallego, and F. Méhats, Entropic discretization of a quantum drift-diffusion model, SIAM J. Numer. Anal., 43 (25), pp [14] P. Degond, C. D. Levermore, and C. Schmeiser, A note on the energytransport limit of the semiconductor Boltzmann equation, in Transport in transition regimes (Minneapolis, MN, 2), vol. 135 of IMA Vol. Math. Appl., Springer, New York, 24, pp [15] P. Degond, F. Méhats, and C. Ringhofer, Quantum hydrodynamic models derived from the entropy principle, Contemp. Math. 371, Amer. Math. Soc., Providence RI (25). [16], Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (25), pp

123 REFERENCES 123 [17] P. Degond and C. Ringhofer, A note on binary quantum collision operators conserving mass, momentum and energy, C. R. Acad. Sci. Paris, 336 (23), pp. Ser I, [18], Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (23), pp [19] P. Degond and C. Schmeiser, Macroscopic models for semiconductor heterostructures, J. Math. Phys., 39 (1998), pp [2] G. Dresselhaus, Spin-orbit coupling effects in zinc blende structures, Phys. Rev., 1 (1955), pp [21] R. J. Elliott, Theory of the effect of spin-orbit coupling on magnetic resonance in some semiconductors, Phys. Rev., 96 (1954), pp [22] A. Fert, A. Friederich, and al, Giant magnetoresistance of (1)fe/(1)cr magnetic superlattices, Phys. Rev. Lett., 61 (1988), pp [23] G. Fishman and G. Lampel, Spin relaxation of photoelectrons in p-type gallium arsenide, Phys. Rev. B, 16 (1977), pp [24] G. Gallego and F. Méhats, On Quantum Hydrodynamic and Quantum Energy Transport Models, Commun. Math. Sci., 5 (27), pp [25] I. Gasser and R. Natalini, The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors, Quart. Appl. Math., 57 (1999), pp [26] F. Golse and D. Levermore, Stokes-Fourier and acoustic limits for the Boltzmann equations : Convergence proofs, Comm. Pure Appl. Math., 55 (22), pp [27] F. Golse and F. Poupaud, Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, Asymptotic Anal., 6 (1992), pp [28] F. Golse and L. Saint-Raymond, Hydrodynamics limit for the Boltzmann equation, Lecture Porto Ercole, (24). [29], The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent Math., 155 (24), pp [3] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon Press, Oxford, [31] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), pp

124 124 REFERENCES [32] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 199. [33] M. S. Mock, Analysis of mathematical models of semiconductor devices, vol. 3 of Advances in Numerical Computation Series, Boole Press, Dún Laoghaire, [34] A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, [35] Y. Pershin, S. Saikin, and V. Privman, Semiclassical transport models for semiconductor spintronics, Electrochem. Soc. Proc., (25), pp [36] Y. V. Pershin and V. Privman, Focusing of spin polarization in semiconductors by inhomogeneous doping, Phys. Rev. Lett., 9 (23), p [37] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation : analysis of boundary layers, Asymptotic Anal., 4 (1991), pp [38] Y. Qi and S. Zhang, Spin diffusion at finite electric and magnetic fields, Phys. Rev. B, 67 (23), p [39] Y. P. Raizer, Gas Discharge Physics, Springer, Berlin, [4] M. Reed and B. Simon, Methods of modern mathematical physics., Academic Press, New York, second ed., Functional analysis. [41] S. Saikin, A drift-diffusion model for spin-polarized transport in a twodimensional non-degenerate electron gas controlled by spin orbit interaction, Journal of Physics : Condensed Matter, 16 (24), pp [42] S. Saikin, Y. Pershin, and V. Privman, Modelling for semiconductor spintronics, IEE Proceedings Circuits, Devices and Systems, 152 (25), pp [43] C. Schmeiser and A. Zwirchmayr, Elastic and drift-diffusion limits of electron-phonon interaction in semiconductors, Math. Models Methods Appl. Sci., 8 (1998), pp [44] T. Valet and A. Fert, Theory of the perpendicular magnetoresistance in magnetic multilayers, Phys. Rev. B, 48 (1993), pp [45] C. Villani, Limite hydrodynamiques de l équation de Boltzmann, Séminaire Bourbaki, 53ème année, no 893 (2 21). [46] Z. G. Yu and M. E. Flatté, Electric-field dependent spin diffusion and spin injection into semiconductors, Phys. Rev. B, 66 (22), p

125 REFERENCES 125 [47] Z. G. Yu and M. E. Flatté, Spin diffusion and injection in semiconductor structures : electric field effects, Phys. Rev. B, 66 (22), p [48] I. Zutic, J. Fabian, and S. Das Sarma, Spin-Polarized Transport in Inhomogeneous Magnetic Semiconductors : Theory of Magnetic/Nonmagnetic p n Junctions, Phys. Rev. Lett., 88 (22), p [49] I. Zutic, J. Fabian, and S. D. Sarma, Spintronics : Fundamentals and applications, Reviews of Modern Physics, 76 (24), p. 323.

126

127 Chapter 3 Numerical Applications Abstract The purpose of this chapter is to somewhere validate numerically the models derived in the previous chapters. Two numerical applications are carried out. The first one concerns the simulation of transistor with spin rotation effect (spin-fet). Following the work of [6, 22], a subband drift-diffusion Schrödinger-Poisson model with spin rotation and relaxation effects is derived and used for the simulation. The second application is a well known effect in semiconductor spintronics : the accumulation of spin polarization density at the interface between two semi-conductor regions with different doping levels [18, 17]. This phenomena is illustrated using two-component drift-diffusion Poisson model. Moreover, we show the action of the Rashba spin-orbit coupling on this spin accumulation effect by means of spin vector drift-diffusion Poisson model. 127

128 128 CHAPTER 3. NUMERICAL APPLICATIONS 3.1 Modelling and numerical implementation of spin-fet Introduction Recently, the spin-related properties of charge carries in semiconductors have generated a huge attention leading to proposition of new magnetoelectronics devices. The Rashba term in the spin-orbit coupling [11] appearing in a quantum well under electric field along the growth direction makes feasible the control of electron spin orientation through the confining electric field. This mechanism leads to the concept of the spin Field Effect Transistor, the spin-fet [14]. The spin-fet is a High Mobility Transistor (HEMT) in which the source and drain contacts are ferromagnetics. A HEMT transistor is composed of a channel which is an active region connected to two electrodes called the source and drain. It is based on the formation of electron confined gas in the channel at the heterointerface between different semiconductor heterostructures. The confinement in the channel is commanded by a Schottky gate. The source and drain are high doped regions and play the role of two small reservoirs. Apply at the electrodes a drain-source potential, V DS, a drain current can be established. The idea of spin-fet, due to Datta and Das [14], consists in replacing the source and drain by ferromagnetic contacts. The source then acts as polarizer and injects a spin polarized current. In the channel, the control of electron spin orientation is possible by the applied gate potential through the Rashba spin-orbit coupling. The drain plays the role of selector. It collects preferentially electrons with spin orientation depending on its magnetic moment. The device considered here is a double gate MOSFET represented by Figure 3.1. We do not take a ferromagnetic contacts at the source and drain. We assume that a spin polarized current is injected from the source to the channel. The object is to show numerically the action of the gate voltage (via the Rashba coupling) on the spin dynamics in 1DEG. To this aim, we take a two dimension device in the (xz) plane. We assume that the particles are confined in the z direction and the transport is allowed in the x direction. The Rashba spin precession vector is then oriented in the y direction (perpendicular to device plane (xz)) and does not change direction. The aim of the simulation is to illustrate the efficiency of the gate control of the spin orientation in this case. The results presented here are well known in semiconductor spintronics. See for example [8] for the study of 1DEG. However, when taking a two dimensional channel with width equal to W in the y direction, the situation is more complicated. Indeed, in two-dimensional electron gas, the Rashba effective field direction depends

129 3.1. MODELLING AND NUMERICAL IMPLEMENTATION OF SPIN-FET 129 strongly on v (or on the wave vector). Then, the electron/crystal scattering randomize the Rashba effective field direction. So, during the motion of one electron, its spin orientation becomes progressively incoherent. The spin polarization of the electron population may be thus fast relaxed (by the D yakonov-perel relaxation mechanism). Using Monte Carlo technique, A. Bournel and al study the gate induced spin precession in 2DEG [8, 9, 1]. They studied the influence of the width W on the spin diffusion length L s or spin relaxation length induced by the interactions. It is shown that the D yakonov-perel spin relaxation mechanism tends to vanish (or to disappear as in the 1DEG case) when the channel width decreases. They prove that it is sufficient to take W less than 1 nm at the room temperature to have a spin relaxation length exceeding 1µm. In this case, one obtains an efficient gate-controlled spin precession in the channel as in the one dimensional case presented here. For this, the spin orientation injected by the source contact has to be perpendicular to the source/channel interface. We refer also to [2, 21] for the study of gate-induced spin precession in 2DEGs. We don t study here the 2D channel case. The study of spin transport properties in 2DEG will be the subject of future works. The model used for the simulation is a coupled quantum classical model. In the confinement direction, the length scale is of the order of the electron de Broglie wavelength and a quantum models have to be adapted in this direction. However the length scale in the transport direction is much bigger and a classical transport model can be used. This gives rise to the subband theory [1, 2, 15]. The subband models are the subject of many recent works. In [3], as mentioned in the first chapter of this thesis, a partially quantum/kinetic subband model is derived by a partially semi-classical limit from a full quantum model. A rigorous analysis of the limit model (Vlasov Schrödinger-Poisson model) is then investigated in [4]. We refer also to [7, 22, 5], for the derivation and study of quantum/fluid type models. The next subsection is dedicated to give more precise description on the subband quantum/drift-diffusion model A coupled quantum/drift diffusion model with spinorbit effect In this part, we will derive a macroscopic subband model by performing a diffusion limit of the Boltzmann Schrödinger system with spin-orbit term. We adapt the work of [22]. Assume that the quantum effects take place in one direction of the space denoted by z, z [, 1], and in the other directions, x R d, the particles move classically. At a time t > and a position (x, z) R d [, 1], the density of the system is given by

130 13 CHAPTER 3. NUMERICAL APPLICATIONS N ε (t, x, z) = Fn(t, n 1( ε x, v)dv) χ n (t, x, z) 2. (3.1.1) R d The unknowns of the problem are (ɛ n, χ n, F ε n) n 1. The subbands are characterized by the complete set, (ɛ n, χ n ) n, of the eigenelements of the one dimensional Schrödinger d 2 operator 1 + V with homogeneous Dirichlet data. More precisely, the sequence 2 dz2 (ɛ n (t, x), χ n (t, x,.)) n verifies 1 2 d 2 dz 2 χ n + V χ n = ɛ n χ n, χ n (t, x,.) H 1 (, 1), 1 χ n χ m = δ nm. (3.1.2) The potential V (t, x, z) is supposed to be given and regular function (see Assumption 3.1.2). Generally, V is not given but satisfies the Poisson equation if one accounts the electrostatic forces. For each subband, the distribution function, F ε n(t, x, v) H + 2 (C), satisfies the following scaled spinor Boltzmann equation t F ε n + 1 ε (v. xf ε n x ɛ n. v F ε n) = 1 ε 2 Q(F ε ) n + i 2 [Ωε n. σ, F ε n], (3.1.3) where Ω ε n(t, x, v) = 1 ε Ωo n(t, x, v) + Ω e n(t, x, v) (3.1.4) such that Ω o n, Ω e n are respectively odd and even functions with respect to v satisfying Assumption Here, the collision operator takes the form Q(F ) n = n R 2 σ nn (v, v ){M n (v)f n (v ) M n (v )F n (v)}dv (3.1.5) which accounts the transition between the subbands. The normalized Maxwellian M n is given by M n (v) = where Z is the repartition function 1 (2π) d 2 Z e ( 1 2 v 2 +ɛ n) (3.1.6) Z(t, x) = n e ɛn(t,x). (3.1.7) Initially, we will take F ε n(, x, v) = F in,n (x, v). (3.1.8) We fix as in the previous chapter the following assumptions on the cross-section, the potential and the effective field.

131 3.1. MODELLING AND NUMERICAL IMPLEMENTATION OF SPIN-FET 131 Assumption The cross-section is assumed to be symmetric and bounded from above and below. Namely, α 1, α 2 >, < α 1 σ nn (v, v ) α 2, n, n 1, v R d, v R d. Assumption For any fixed T >, (t, x, z) V (t, x, z) is a real non negative function belonging to C 1 ([, T ], W 1, (R d (, 1))). Assumption For any T >, (Ω o n) n N, (Ω e n) n N C 1 ([, T ] R d x R d v, R 3 ) are two sequences of respectively odd and even functions. In addition, there exists (C n ) n l 1 (R + ) and m N such that Ω o(e) n + η {x i,v i } We will denote by l 1 (L 2 M ) the space given by l 1 (L 2 M) = { (F n ) n 1 L 2 M / η Ω o(e) n C n (1 + v ) m. (3.1.9) n 1 F n 2 2 dv < + R M d n It is simple to show that the collision operator (3.1.5) is a linear, self-adjoint and nonpositive operator on l 1 (L 2 M ) satisfying 1. n 1 Q(F ) R d n (v)dv =, 2. Ker(Q) = {F l 1 (L 2 M ), such that N H+ 2 (C), F n (v) = NM n (v), n 1}, { 3. R(Q) = Ker(Q) = F = (F n ) n l 1 (L 2 M ), such that ( )} n R F d n (v)dv =, 4. n 1 Q(F ) n, F n M α 1 n 1 F n P(F ) n M, when P is the orthogonale projection on Ker(Q). The matrices, (D i ) i=1..4, appearing in the general drift diffusion model derived in Section 6 of the previous chapter are defined in this case as follows. Proposition Let D 1, D 2, D 3 and D 4 be the matrices defined respectively by D 1 = (θ 1,n (v) v)dv M dd (R), D 2 = (v θ 2,n (v))dv M d3 (R), n R d n R d D 3 = (Ω o n(v) θ 1,n (v))dv M 3d (R), D 4 = (θ 2,n (v) Ω o n(v))dv M 33 (R) n R d n R d (3.1.1) where θ 1, θ 2 are respectively the unique solutions of Q(θ 1 Id) n = vm n (v)id, θ 1,n (v)dv =, (3.1.11) R d Q(θ 2 Id) n = Ω o nm n (v)id, n n } R d θ 2,n (v)dv =. (3.1.12) Then, The matrices D 1 and D 4 are symmetric positive definite and t D 3 = D 2..

132 132 CHAPTER 3. NUMERICAL APPLICATIONS Passing to the limit ε, we get an hybrid Drift-Diffusion Schrödinger system. Namely, applying the same ideas as in the previous chapter, the following results hold. Theorem Let T >, (F in,n ) n l 1 (L 2 M (dxdv)) and assume that Assumptions 3.1.1, and hold. Then, for all ε >, the model (3.1.3)...(3.1.8) admits a unique weak solution (Fn) ε n C ([, T ]; l 1 (L 2 M (dxdv))). In addition, there exists N s L 2 ([, T ] R d, H 2 + (C)) such that Fn ε N s(t, x) Z(t, x) M n(t, x, v) weak in L ([, T ], l 1 (L 2 M (dxdv))) and the surface density N s ε := Fn(t, ε x, v)dv n 1 R d converges weakly to N s in L 2 ([, T ] R d, H 2 + (C)). If we decompose the density matrix N s into a spin independent and a spin dependent parts : N s = n c 2 I 2 + n s. σ then, the charge density n c and the spin density n s satisfy t n c div x (D 1 ( x n c + x V s n c )) =, t n s div x (D 1 ( x n s + x V s n s ) + 2(D k 2 n s ) k=1,2,3 ) = N s (, x) = N in := n 1 R d F in,n (x, v)dv, with Ω = div x D 2 D 3 ( x V s ) + H e, H e (t, x) = n The potential V s is given by Ω n s + (D 4 trd 4 )( n s ) 2 n s τ sf, (3.1.13) R d Ω e n(t, x, v)m n (t, x, v)dv. (3.1.14) V s = Log( n e ɛ n ). Furthermore, the model (3.1.13) satisfies the maximum principle. i.e. if initially N in L 2 (R d, H + 2 (C)) then, N s (t, x) is an Hermitian positive matrix for all t [, T ] and x R d. Remark The total density N ε (t, x, z) given by (4.5.3) converges weakly to N in L 2 ([, T ] R d x [, 1] z, H + 2 (C)) such that and N s (t, x) = 1 N(t, x, z) = N s(t, x) Z(t, x) e ɛn(t,x) χ n (t, x, z) 2 n N(t, x, z)dz is the surface density satisfying (3.1.13).

133 3.1. MODELLING AND NUMERICAL IMPLEMENTATION OF SPIN-FET Setting of the problem and numerical results For the application, we will take the Rashba spin-orbit effect. The Rashba effective field in the subband kinetic/quantum model is given in 2DEG by Ω o n = α n (t, x)( v 2, v 1, ), (3.1.15) where α n (t, x) = 1 z V (t, x, z) χ n (t, x, z) 2 dz. (3.1.16) See the first chapter. In the example considered here, the transport takes place in one dimension and the subband Rashba spin-orbit effect is then characterized by the following effective field Ω o n(t, x, v) = α n (t, x)ve y, (t, x, v) R + R 2 x,v. (3.1.17) We denote by (e x, e y, e z ) an orthonormal basis of the space with e x and e z are respectively the unit vectors of the transport direction x and the confinement direction z. We suppose in addition, for simplicity, that the cross-section σ n,n (v, v ) is a constant independent of n, n, v, v and let τ > be the relaxation time such that σ nn (v, v ) = 1 τ, for all n, n, v, v. Then, θ 1 and θ 2 can be explicitly computed θ 1,n = τvm n (v), θ 2,n = τω o n(v)m n (v). (3.1.18) In this case, the matrices D 1, D 2, D 3 and D 4 are equal to D 1 = τid, D 2 = t D 3 = τ (v Ω o n(v))m n (v)dv, D 4 = τ n R d n R d (Ω o n Ω o n)m n dv. Thus, for d = 1 and Ω o n given by (3.1.17), we have D 1 = τid, D 3 = t D 2 = τc 1 e y where C 1 = 1 α n e ɛn (3.1.19) Z and 1 D 4 trd 4 = τc 2 where C 2 = 1 α Z ne ɛ 2 n. (3.1.2) n 1 Let us now summarize the complete stationary Drift Diffusion Schrödinger Poisson model written with the physical constants used for the simulation. We assume that the device occupies a 2D domain denoted by [, L] [, l]. At a position (x, z) [, L] [, l], the total density matrix for Boltzmann statistics is N(t, x, z) = N s(t, x) Z(t, x) n e βɛn(t,x) χ n (t, x, z) 2, (3.1.21) n=1

134 134 CHAPTER 3. NUMERICAL APPLICATIONS where β = 1 k B T, k B is the Boltzmann constant and T denotes the temperature. The complete set of eigenfunctions and eigenvalues (χ n,ɛ n ) n 1 satisfies 2 2m d 2 dz 2 χ n + (V + V c )χ n = ɛ n χ n, χ n (t, x,.) H 1 (, l), l χ n χ n = δ nn, (3.1.22) where is the planck constant, m is the effective mass and V c is a given potential barrier between the silicon and the oxyde in the nanotransistor (see Fig. 3.1). The electrostatic potential is given by U = V e satisfies the Poisson equation x,z V = where e is the elementary charge and e ε r ε (tr(n) N D ) (3.1.23) where ε r is the relative permittivity, ε the permittivity constant of the vacuum and N D is the doping density which is equal to N + R + in the drain and the source and N R + in the channel. Writing the surface matrix density, N s, as : N s (t, x) = n c(t, x) I 2 + n s (t, x) σ, n c and n s satisfy the following stationary drift 2 diffusion equations div x (j c ) =, j c = D( x n c + β x V s n c ), div x ( j s ) = D( x C 1 x V s C 1 )e y n s DC 2 n s, (3.1.24) j s = D( x n s + β x V s n s + 2C 1 (e y n s )), where D is the diffusion constant D = µk B T for a constant mobility µ and ( + V s = k B T log e βɛ ) k. (3.1.25) The functions C 1, C 2 are given in (3.1.19) and (3.1.2) and n s is the part of n s in the (ox, oz) plane perpendicular to the Rashba effective field (or to e y ) : k=1 n s = n s ( n s e y )e y. Remark that the total charge density appearing in the right hand side of (3.1.23) is given by tr(n)(t, x, z) = n c(t, x) Z(t, x) e βɛn(t,x) χ n (t, x, z) 2. (3.1.26) n=1 Description of the scheme. To solve numerically the model (3.1.22)-(3.1.26), A finite element method with Gummel iterations is used [16]. We give here a brief

135 3.1. MODELLING AND NUMERICAL IMPLEMENTATION OF SPIN-FET 135 description of the algorithm. We refer to [6, 19] for more details. We mesh the domain by N x N z, (x i, z j ) 1 i Nx,1 j N z, nodes. We begin by resolving the 2D Schrodinger-Poisson system when there is no applied voltage. The boundary conditions used are as follows. At the source and the drain, the potential does not depend on the transport direction, we take V (x =, z) = V (x = L, z) = V b with V b (z) is the solution of 1D Schrödinger-Poisson system in the z direction. The surface density at the source and the drain is taken equal to N + l Si. For z = or z = l, we take a Dirichlet boundary condition on the gate contacts, V (x, z = ) = V (x, z = l) = V gs. When a drain source voltage, V DS, is applied, we resolve the Schrödinger-Poisson system coupled to the charge Drift Diffusion equation (the first equation of (3.1.24)) by Gummel iterations starting from the obtained potential in the first step and incrementing by.2v. For the drift diffusion equation a Dirichlet conditions are used at x = and x = L. For the potential we impose Dirichlet boundary conditions at the Gate and at the source and the Drain contacts : V (x =, z) = V b (z), V (x = L, z) = V b (z) + V DS. The Gummel iterations can be summarized as follows. For a given potential, V old, a diagonalization of the one dimensional Schrödinger operator (3.1.22) gives N x sets of eigenfunctions {χ k (x i, z)} i=1,..nx and eigenvalues {ɛ k (x i )}. The effective potential V s is then computed from (3.1.25). This allows to obtain the surface charge density n c by solving the first drift diffusion equation of (3.1.24) and then to compute a new potential V new thanks to the resolution of (3.1.23) with (3.1.26). We repeat these steps until the difference V new V old L be sufficiently small. Noting that the Poisson equation is resolved by implementing a quasi-newton method [12]. Finally the spin part of the drift diffusion model (3.1.24) is resolved using Dirichlet boundary conditions. We assume that at x = or x = L, the vectors n s (x = ) and n s (x = L) are in the plane of the domain ((xz) plane). Then, the spin vector density n s remains in this plane or it has no component in the y direction (parallel to e y ) and its perpendicular part, n s (in the (xz) plane) solves div x ( j s ) = D(C 2 n s ( x C 1 x V s C 1 )I( n s )), j s = D( x n s + β x V s n s + 2C 1 I( n s )) ( ) 1 with I =, n s R 2 C. We resolve this equation in C, in which I is 1 the pur imaginary unit. Numerical results. The geometry of the device is given by Figure 3.1. We summarize in Table 3.1 the main values used in the simulation. We assume that the electrons are injected into the domain with spin vector density along the x-axis.

136 136 CHAPTER 3. NUMERICAL APPLICATIONS Fig. 3.1 Schematic of the modeled device. Tab. 3.1 Table of the main values used. Paramètre Valeur Longueur Valeur N cm 3 L S 5 nm N 1 21 cm 3 L C 2 nm U c 3 ev L D 5 nm ε r (SiO 2 ) 3.9 L ox 2nm ε r (Si) 11.7 L Si 6 nm

137 3.1. MODELLING AND NUMERICAL IMPLEMENTATION OF SPIN-FET 137 More precisely, we take n s (x = ) = (n c (x = ), ). This means that the injected current in the device is supposed 1% spin polarized. Recall that the density spin polarization in this case is equal to ns(x=) n c(x=) = 1. We present in Figure 3.2 the sinus of the angle between n s and the x-axis for different values of the gate potential V gs and for drain source voltage V DS =.1V. Figure 3.2 shows that the spin orientation rotates in the (xz) plane with an angular frequency proportional to the gate voltage. This shows also that the spin vector orientation reaching the drain contact can be controlled by V gs Vgs=V Vgs=.35 Vgs=.7 Vgs=1.5 Vgs= Vgs=1.75 V Vgs=2.1 V Vgs=2.45 V Vgs=2.8 V Vgs=3.15 V Sin(alpha) sin(alpha) x (nm) x (nm) Fig. 3.2 The sinus of the angle (α) between the the density vector n s different values of the gate potential denoted by V gs and for V DS =.1V. and the x-axis for 4 x 14 4 x 14 Charge current in standard MOSFET (A.m 1 ) Spin drain current in spin FET (A.m 1 ) Vgs Vgs Fig. 3.3 I V gs characteristics for the device for V DS =.1V. At the left, the charge current in a standard HEMT is represented. At the right, we plot the spin current density at the drain in spin-fet with spin selective drain in the direction parallel to x. In Figure 3.3, we plot the current-voltage (I V gs ) characteristics for V DS =.1V.

138 138 CHAPTER 3. NUMERICAL APPLICATIONS In standard MOSFET, the current (or charge current given by the first equation of (3.1.24)) increases with the gate potential (left). The diffusion constant is given by the Einstein relation : D = µk B T where µ =.12m 2 V 1 s 1 and T = 3K. However, if we assume that the drain is a selective contact of electron spin. The magnitude of the drain current depends then on the orientation of the spin of the electrons reaching the drain with respect to the orientation of the magnetic moment of this contact. We consider a drain contact which selects spins parallel to the x direction. The x-part of the spin drain current as function of V gs is illustrated in Figure 3.3 (right). It is not any more an increasing function. It oscillates with respect to the gate voltage V gs. If the spin density reached the drain contact is oriented parallel to the x axis, the drain s current is important. In the other case, the current in the device is weak. 3.2 Spin accumulation in inhomogeneous semiconductor interfaces The accumulation of electron spin polarization density at an interface separating two semiconductor regions with different doping levels is a well known effect in spintronics. We refer to [18, 17, 13] in which this effect was studied using two-component type models. In this section, some numerical results illustrating the propagation of spin polarization density through a boundary between two different doped semiconductor regions will be presented and interpreted. Two kind of transport models are used : two-component and spin vector Drift-Diffusion models coupled with the Poisson equation for self-consistent forces Presentation of the system The geometry of the system under consideration is represented by figure 3.4. We consider in dimension one, two semi-conductor layers with doping densities N 1 and N 2 respectively such that N 2 N 1. Supposing that a spin-polarized electrons are injected in the left semiconductor, under influence of the electric field, the spin polarized current is driven toward the second higher doped region. It was found that, see [18], the electron spin polarization density is accumulated near the interface between the two regions and becomes more pronounced when N 2 increases. To illustrate this

139 3.2. SPIN ACCUMULATION IN INHOMOGENEOUS SEMICONDUCTOR INTERFACES139 Fig. 3.4 Schematic of the system under investigation : spin-polarized electrons are injected in the first semiconductor (S 1, N 1 ) with doping N 1 and transported under the action of the electric field in the x direction toward another N 2 doped semiconductor region (S 2, N 2 ). phenomena, we begin by using a self consistent two-component drift-diffusion model t n ( ) div x j ( ) = n ( ) n ( ), ( τ sf j ( ) = D x n ( ) + e ) k B T xv n ( ), (3.2.1) x V = e (n N i ). ε r ε We assume that the system lies on the interval I = [ 1x, 1x ] with x is the scale of the transport direction x. For the numerical simulation we will take x = m in the sequel. The boundary between the two regions is supposed at point b I. The spin-up and spin-down densities, n (t, x) and n (t, x), satisfy the 2-component drift-diffusion system (3.2.1) with spin-flip collision term. The constant τ sf is the spin-flip relaxation time and D is the diffusion constant. We denote by V the electrostatic potential satisfying the Poisson equation in which n(t, x) = n (t, x)+n (t, x) is the total density of the particles (or the charge density) and N i (x) is the doping function N i (x) = Numerical results { N 1 if x < b, N 2 if x > b. (3.2.2) We have solved numerically system (3.2.1) using finite elements method with Gummel iterations and taking Dirichlet boundary conditions for the potential and the densities. The results presented here are representative for the general idea of

140 14 CHAPTER 3. NUMERICAL APPLICATIONS 16 x Electric field (V/m) x (m) x 1 6 Fig. 3.5 Electric field profil near the boundary, E = x V, for N 2 N 1 = 5. the spin accumulation in inhomogeneous semiconductor interfaces. The values used for the different parameters are : x = m, b =.8 1 7, N 1 = 1 21 m 3, T = 3K. Figure 3.5 shows the electric field profile (E = x V ) near the boundary in the case when N 2 = 5N 1. A peak is formed at the boundary due to the electrons diffusing from the highly doped right region to the less doped left region. Let us now present the evolution of the spin polarization density p(t, x) defined by p(t, x) = n (t, x) n (t, x). Figure 3.6 shows the evolution of p(t, x) for different values of N 2 with respect to N 1. The initial spin-polarization (at t = ) is represented by the dashed lines. In each of the different cases presented, p(t, x) is plotted for different instances ; it converges when t increases to an equilibrium profile. In the case N 2 = N 1, no accumulation of spin-polarization is present at the boundary and p decreases exponentially to zero. However, when N 2 is greater than N 1 (we give in Figure 3.6 three cases : N 2 = 5N 1, N 2 = 1N 1 and N 2 = 2N 1 ), we see that a peak of spin polarization is formed near the boundary in the right region and becomes more pronounced when N 2 increases Spin accumulation and Rashba spin-orbit effect The aim of this subsection is to study numerically the effect of the spin-orbit interaction on the spin accumulation in inhomogeneous semiconductor interfaces.

141 3.2. SPIN ACCUMULATION IN INHOMOGENEOUS SEMICONDUCTOR INTERFACES141 1 x 12 1 x N 2 =N 1 8 N 2 = 5 N n n (m 3 ) n n (m 3 ) x (m) x x (m) x x 12 1 x N 2 = 1 N 1 8 N 2 = 2 N n n (m 3 ) n n (m 3 ) x (m) x x (m) x 1 6 Fig. 3.6 The evolution of the spin-polarization density for different values of N 2 with respect to N 1. In each case, the dashed line is the initial spin-polarization profile (at t = ). The continuum curves represent the spin-polarization density for different instances.

142 142 CHAPTER 3. NUMERICAL APPLICATIONS We will use the spin vector drift-diffusion model derived in the previous chapter coupled with the Poisson equation. Let us write this model in the case of relaxation time approximation and with one dimensional Rashba spin-orbit effect where the effective field is given by Ω = αve y such that α is the order of the Rashba interaction and e y is the unit vector of the y direction (perpendicular to the transport direction x). This effective field can appear for example if an electric field is applied along the z direction. The model is the following t n c div x (D( x n c + β x V n c )) =, t Ns div x (D( x Ns + β x V N s + 2αe y N s )) = αβd x V (e y N s ) α 2 DN s x V = e (n c N i ), ε r ε 2 N s τ sf, (3.2.3) where β = e k B T and D is the diffusion constant. We recall that n c is the charge density and N s is the spin-vector density where N s represents the polarization density (or n n in the 2-component description). We denoted in the second equation by N s the perpendicular part of N s with respect to the effective field direction (or e y direction). We have plotted in figure 3.7 the polarization density profile (i.e. the equilibrium profile of N(t, x) when t increases) for different values of the spinorbit coefficient (α). Figure 3.7 shows the variation of the spin accumulation at the boundary with respect to α. Let us mentioned the presence of oscillation effect of the spin accumulation due to the rotational and relaxation effects induced by the Rashba interactions.

143 REFERENCES 143 Fig. 3.7 The polarization profile as a function of the transport direction x and the coefficient of the Rashba spin-orbit coupling α. References [1] T. Ando, A. B. Fowler, and F. Stern, Electronic properties of twodimensional electron systems, Rev. Mod. Phys., 54 (1982), p [2] G. Bastard, Wave mechanics applied to semiconductor heterostructures, Les éditions de Physique, [3] N. Ben Abdallah and F. Méhats, Semiclassical analysis of the Schrödinger equation with a partially confining potential, J. Math. Pures Appl. (9), 84 (25), pp [4] N. Ben Abdallah and F. Méhats, On a Vlasov-Schrödinger-Poisson Model, Comm. Partial Differential Equations, 29 (24), pp [5] N. Ben Abdallah, F. Méhats, and C. Negulescu, Adiabatic quantumfluid transport models, Comm. Math. Sci., 4 (26), pp [6] N. Ben Abdallah, F. Méhats, P. Pietra, and N. Vauchelet, A driftdiffusion subband model for the simulation of the Double-Gate MOSFET, IEEE- NANO 25, Conference on Nanotechnology, Nagoya. [7] N. Ben Abdallah, F. Méhats, and N. Vauchelet, Analysis of a Drift-Diffusion-Schrödinger-Poisson model, C. R. Acad. Sci. Paris, Ser. I 335, pp [8] A. Bournel, Magnéto-électronique dans des dispositifs à semiconducteurs, Ann. Phys. Fr, 25 (2), pp